Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Proving that $\frac{1}{n^2} - \frac{1}{(n+1)^2} \approx \frac{2}{n^3}$ when $n$ is very large This is an example from Mathematical Methods in the Physical Sciences, 3e, by Mary L. Boas.
My question is,
\begin{equation}
\frac{1}{n^2} - \frac{1}{(n+1)^2} \approx \frac{2}{n^3}
\end{equation}
can also be written as,
\begin... | The claim can be shown with straightforward algebra as b00n heT points out. However, you could also use the mean value theorem, which formalizes your idea.
As for your approach, it's not clear to me why you wrote $\frac{1}{n^2}=\frac{1}{(-n)^2}.$ This is of course true, but this does not make $n=-n$, which is essential... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4379373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
How many necklaces can be formed with $6$ identical diamonds and $3$ identical pearls
Find number of ways to make a necklace (or a garland) consisting of $6$ identical diamonds and $3$ identical pearls.
I got the correct answer $7$ by taking different cases but when I applied the formula for arrangement, i.e. $$\frac... | There are two possibilities here, rotational symmetry (necklace) or
dihedral symmetry (bracelet). This is the OEIS naming convention. For the
first one we have the cycle index of the cyclic group:
$$Z(C_n) = \frac{1}{n} \sum_{d|n} \varphi(d) a_d^{n/d}.$$
For second one we have the cycle index of the dihedral grou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4379991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
A typical inequality: $\frac{8}{27}\leq\frac{(x^2+yz)(y^2+zx)(z^2+xy)}{(xy+yz+zx)^3}$
For $x, y, z\in (0, \infty)$ prove that: $$\frac{8}{27}\leq\frac{(x^2+yz)(y^2+zx)(z^2+xy)}{(xy+yz+zx)^3}.$$
My attempts to apply media inequality or other inequalities have been unsuccessful. In desperation I did the calculations an... | Add the following inequalities:
(1) Schur's inequality times $2xyz$:
$$xyz(x(x-y)(x-z)+y(y-z)(y-x)+z(z-x)(z-y))\ge 0$$
That is
$$2(x^4yz+y^4zx+z^4xy)+6x^2y^2z^2\ge 2(x^3y^2z+x^3yz^2+x^2y^3z+xy^3z^2+x^2yz^3+xy^2z^3)$$
(2) $12.5$ times $x^3+y^3-x^2y-xy^2\ge 0$: that is $12.5\sum_{cyc}xyz(x^3+y^3-x^2y-xy^2)\ge 0$. That is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4381204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Evaluate $\int_{0}^{\infty} \frac{\ln x}{\left(x^{2}+1\right)^{n}} d x$. Latest Edit
By the contributions of the writers, we finally get the closed form for the integral as:
$$\boxed{\int_{0}^{\infty} \frac{\ln x}{(x^{2}+1)^n} d x =-\frac{\pi(2 n-3) ! !}{2^{n}(n-1) !} \sum_{j=1}^{n-1} \frac{1}{2j-1}}$$
I first evaluat... | Your integral can be expressed as the derivative of the complete beta function:
From Fubini-Tonelli we can interchange limit/derviative and integral:
$$I = \int_{0}^{\infty} \frac{\ln x}{\left(x^{2}+1\right)^{n}} d x = \int_{0}^{\infty} \frac{\left(\frac{d}{dt}\Big|_{t=0+} x^t\right)}{(x^2+1)^n}dx = \frac{d}{dt}\Big|_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4382586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 4
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$\text { Show that } \sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)=-2 \pi+2 \cos ^{-1} x \text { if }-1 \leq x \leq-\frac{1}{\sqrt{2}} $
Show that
$$\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)=-2 \pi+2 \cos ^{-1} x$$ if $-1 \leq x \leq-\frac{1}{\sqrt{2}}.$
I tried solving this question a lot but I’m unable to. My answer co... | Continuing from the substitution in the question, where OP put $x=\cos \theta$. Here I pick $\theta$ to be the particular value $\frac{3\pi}4\le \theta\le \pi$, so that $\theta$ is the principal value $\theta = \cos^{-1}x$.
Using the formula to find general solutions of inverse $\sin$, if given $\sin 2\theta$, $2\theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4384502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
${ n \choose k}={ n-1 \choose k-1}+{ n-2 \choose k-1}+{ n-3 \choose k-1}+...+{ k-1 \choose k-1}$ i tried to prove this by using the concept of composition
${ n \choose k}={ n-1 \choose k-1}+{ n-2 \choose k-1}+{ n-3 \choose k-1}+...+{ k-1 \choose k-1}$
my attempt:
for explain my idea let's take $ n=6$ and $k =4$
so ${ 6... | Yes, your proof is valid. In words, you counted the number of solutions of the equation
$$x_1 + x_2 + \cdots + x_{k + 1} = n + 1 \tag{1}$$
in two different ways. The left-hand side (LHS) counts the number of solutions directly. The right-hand side (RHS) counts the same thing, depending on the value of $x_{k + 1}$ si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4387577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What is the sum of natural numbers that have $5$ digits or less, and all of the digits are distinct? $1+2+3+\dots+7+8+9+10+12+13+\dots+96+97+98+102+103+104+\dots+985+986+987+1023+1024+1025+\dots+9874+9875+9876+10234+10235+10236+\dots+98763+98764+98765=$
The only thing I can do is to evaluate a (bad) upper bound by eval... | Allow $0$ as a lead digit, and regard $025$ as different from $25$. Then the average of each digit is $4.5$. Finally, remove numbers whose first digit is $0$, and whose other digits have average $5$.
So there are $10$ one-digit numbers, average $4.5$, $90$ two-digit numbers, average $4.5 ×11$, $720$ three-digit and s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4395079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Elementary inequalities exercise - how to 'spot' the right sum of squares? I have been working through CJ Bradley's Introduction to Inequalities with a high school student and have been at loss to see how one could stumble upon the solution given for Q3 in Exercise 2c.
Question:
If $ad-bc=1$ prove that $a^2+b^2+c^2+d^2... |
$a^2+b^2+c^2+d^2+ac+bd -\sqrt{3}(ad-bc) \geq 0 \tag{1}$
Completing the squares is made easier by introducing a factor of $\,2\,$, so
let $\,\lambda = \frac{1}{2}\,$ and $\,\mu = \frac{\sqrt{3}}{2}\,$, then the LHS of $(1)$ can be written as:
$$
a^2+b^2+c^2+d^2+ 2\lambda(ac+bd) - 2\mu(ad-bc)
$$
$$\begin{align}
= \;\;\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4396645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 3
} |
Existence of limit $\lim_{n\to\infty}\frac{n!}{n^{n+1/2}e^{-n}}$ It is shown that
$$
e^{\frac{11}{12}}n^{n+1/2}e^{-n}< n!<en^{n+1/2}e^{-n}.
$$
The question is, how to conclude from here that the limit
$$
\lim_{n\to\infty}\frac{n!}{n^{n+1/2}e^{-n}}
$$
exists? I suspect that $\frac{n!}{n^{n+1/2}e^{-n}}$ is decreasing, b... | You can prove the monotonicity as follows. Let $a_n = \frac{{n!}}{{n^{n + 1/2} e^{ - n} }}$. Then
$$
\frac{{a_n }}{{a_{n + 1} }} = \left( {1 + \frac{1}{n}} \right)^{n + 1/2} \frac{1}{e} = \exp \left( {\left( {n + \frac{1}{2}} \right)\log \left( {1 + \frac{1}{n}} \right) - 1} \right).
$$
But
\begin{align*}
\left( {n + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4397778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Inequality with real exponents I had the following problem on a past homework asignment:
For $x,p \in \mathbb{R}^+$
$$\frac{p}{(1+x)^{p+1}}<\frac{1}{x^p}-\frac{1}{(1+x)^{p}}$$
So, I think one way to show this for $1\leq p$ would be with rules for real exponents for real exponents (since $x^{-1} >-1$):
$$(1+\frac{1}{x})... | Case $0 < p < 1$:
The inequality is written as
$$\frac{1 + x + p}{1 + x} \left(1 - \frac{1}{1 + x}\right)^p < 1.$$
Using Bernoulli inequality
$(1 - u)^r \le 1 - ru$ for all $0 < u < 1$ and $0 < r \le 1$, we have
$$\left(1 - \frac{1}{1 + x}\right)^p \le 1 - \frac{1}{1 + x}p.$$
It suffices to prove that
$$\frac{1 + x + p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4398400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $\lim_{n→\infty}\frac{\sqrt{n^2+a^2}}{n}=1$ using "$\varepsilon \to N$" definition how to prove this equation with the "$\varepsilon \to N$" definition? I feel have some trouble.Thanks!
$\lim_{n \to\infty}\frac{\sqrt{n^2+a^2}}{n}=1\\$
I complete my solution:
when $a=0$ it's obviously true. When
$a \neq 0... | welcome to MSE.
another point of view may help.
Taylor expansion for $$\sqrt{1+x}=1+\frac{x}2-\frac{x^2}{8}+o(x^3)$$and
$$\frac{\sqrt{n^2+a^2}}{n}=\frac{\sqrt{n^2+a^2}}{\sqrt {n^2}}=\sqrt{1+(\frac{a}{n})^2}\\=1+\frac 12(\frac{a}{n})^2 -\frac 18(\frac{a}{n})^4+...\\\geq 1+\frac 12(\frac{a}{n})^2$$ so
$$|\frac{\sqrt{n^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4400204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Derivative of $f(x) = \csc(x)\cot(x)$ using first principles. How to find derivative of $f(x) = \csc(x)\cot(x)$ using First principle of derivative?
I tried the following method.
$f(x) = \csc(x)\cot(x) = \dfrac{\cos(x)}{\sin^2(x)}$
Now using the limit,
$f'(x) = \lim\limits_{h\to0}\dfrac{\dfrac{\cos(x+h)}{\sin^2(x+h)} ... | If $$f(x) = \csc(x)\cot(x)\\= \dfrac{\cos(x)}{\sin^2(x)},$$ then
$$f'(x) = \lim_{h\to0}\dfrac{\dfrac{\cos(x+h)}{\sin^2(x+h)} - \dfrac{\cos(x)}{\sin^2(x)}}{h}\\
= \lim_{h\to0}\dfrac{\cos(x+h)\sin^2(x) - \cos(x) \sin^2(x+h)}{h\sin^2(x+h)\sin^2(x)}\\
= \lim_{h\to0}\left(\dfrac{\cos(x+h) -\cos(x)}{h}\frac{\sin^2(x)}{\sin^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4416443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Effective method to solve $\frac{\sqrt[3]{1+x} -\sqrt[3]{1-x}}{\sqrt[3]{1+x} +\sqrt[3]{1-x}} = \frac{x(x^2+3)}{3x^2+1} $ I want to know, is there an easy method to solve below equation $$\frac{\sqrt[3]{1+x} -\sqrt[3]{1-x}}{\sqrt[3]{1+x} +\sqrt[3]{1-x}} = \frac{x(x^2+3)}{3x^2+1} $$ I tried it by plotting and find the so... | I think about it again , $$f(x)=\frac{(\sqrt[3]{1+x} -\sqrt[3]{1-x})}{(\sqrt[3]{1+x} +\sqrt[3]{1-x})}=\\\frac{(\sqrt[3]{\frac{1+x}{1-x}} -1)}{(\sqrt[3]\frac{1+x}{1-x} +1)}$$so $$y=\frac{(\sqrt[3]{\frac{1+x}{1-x}} -1)}{(\sqrt[3]\frac{1+x}{1-x} +1)}=\frac{a+1}{a-1}\\\to a=\frac{y+1}{y-1}\\\to \sqrt[3]{\frac{1+x}{1-x}}=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4417867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Trigonometric and exponential integral $\int _0^{\pi }\frac{\cos \left(a\sin x\right)}{1+a\cos x}e^{a\cos x}dx$ How can we prove this Integral relation?
$$\int _0^{\pi }\frac{\cos \left(a\sin x\right)}{1+a\cos x}e^{a\cos x}dx=\frac{\pi }{e}\cdot \frac{e^{\sqrt{1-a^2}}}{\sqrt{1-a^2}}$$
where $\text{ }a\in(-1,1)$.
| $$I(a)=\int _0^{\pi }\frac{\cos \left(a\sin x\right)}{1+a\cos x}e^{a\cos x}dx=\frac{1}{2}\Re\int _{-\pi}^{\pi }\frac{e^{i \left(a\sin x\right)}}{1+a\cos x}e^{a\cos x}dx=\frac{1}{2}\Re\int _{-\pi}^{\pi }\frac{e^{ ae^{ix}}}{1+a\cos x}dx$$
$$=\Re\int _{-\pi}^{\pi }\frac{e^{ ae^{ix}}e^{ix}}{ae^{2ix}+2e^{ix}+a}dx=\Re\,(-i)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4418256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is it hard to tackle the integral $\int_{0}^{\infty} \frac{x^{2}}{\left(1+x^{4}\right)^{2}} d x?$ Putting $ \displaystyle=4, \alpha=2, n=2 $ in my post,
$$\int_{0}^{\infty} \frac{x^{n}}{\left(1+x^{m}\right)^{\alpha}}dx=\frac{\pi}{m(\alpha-1) !} \csc\frac{(n+1) \pi}{m}\prod_{k=1}^{\alpha-1}\left(\alpha-k-\frac{n+1}{m}\r... | Integrate both sides of the equation $\left(\frac{x^3}{1+x^4}\right)’= \frac{4x^2}{(1+x^4)^2}-\frac{x^2}{1+x^4}$
to obtain
$$\int_{0}^{\infty} \frac{x^{2}}{(1+x^{4})^{2}} d x
= \frac14\int_{0}^{\infty}
\frac{x^{2}}{1+x^{4}} d x=\frac14 \cdot \frac\pi{2\sqrt2}=\frac\pi{8\sqrt2}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4419223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solve $y’’ – 4y’ + 5y = 4e^{2x}\sin(x)$ using $\mathcal D$ operator Hello – I am working through the following question and get stuck at step 6. Could someone please advise in simple terms which I can hopefully understand. Thanks
$$y'' – 4y' + 5y = 4e^{2x}\sin(x)$$
Step one – Order equation so that differential operato... | $e^{ix}=\cos(x)+i\sin(x)$
$\sin(x)= Im(e^{ix})$
$$
{1 \over D^2 + 1} \ \sin x = {1 \over D^2 + 1} \ \ Im(e^{ix})=Im{1 \over D^2 + 1} \ \ e^{ix}=Im{1 \over 2D} \ \ xe^{ix} =x Im{1 \over 2i} \ \ e^{ix}=x Im{1 \over 2i} \ \ (\cos(x)+i\sin(x))=\frac{x}{2} Im\ \ (-i\cos(x)+\sin(x))
= \left[ -{x \over 2} \cos x \right]
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4419803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
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Find all non-similar solutions of matrix equation
Find all unique $($not conjugate to each other by an element of $GL_2(\mathbb{Z}))$ matrices $A \in M_2(\mathbb{Z})$, such that $A ^ 2 - 4A - I = 0$, where $I$ is the identity matrix.
From the equation it can be easily seen that matrix is a solution iff it has trace e... | Your guess is correct. Here is an elementary proof. For $b\neq 0$, let
$$
F(a,b)=\begin{pmatrix} a && b \\ -\frac{a^2-4a-1}{b} && 4-a \end{pmatrix} \tag{1}
$$
Then the initial matrix $A$ is necessarily of the form $F(a,b)$ for some integers $a$ and $b$ (such that $b$ divides $a^2-4a-1$). All the conjugates of $A$ will ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4421127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Does the following definite integral exist? I encounter a problem in which I would need to deal with the folloing definite integral
$$I(t)=\int_{0}^{\infty} \mathrm dp \frac{p^2}{\omega^5} \sin^2\left(\frac{\omega t}{2}\right)$$
in which $$\omega=\sqrt{m^2+p^2}$$
where m is just some positive constant, t is also a pa... | I prefer to add another answer since based on a different approach.
$$I=\int_0^{\infty } \frac{p^2 \sin ^2\left(\frac{t}{2} \sqrt{m^2+p^2} \right)}{\left(m^2+p^2\right)^{5/2}} \, dp=\frac12\int_0^{\infty } \frac{p^2}{ \left(m^2+p^2\right)^{5/2}}\, dp-\frac12\int_0^{\infty } \frac{p^2 \cos \left(t\sqrt{m^2+p^2} \right)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4423139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How to find the inverse function of the function $f$, defined as $f(x)=\frac{x}{1-x^2}$ for $-1 \lt x \lt 1$ I want to find the inverse function of the function $f$, defined as $f(x)=\frac{x}{1-x^2}$ for $-1 \lt x \lt 1$.
Letting $y=f(x)$, we first note that $x=0 \iff y=0$. Next, we consider the case when $y \neq 0$:
... | To find the inverse function, you have to solve for $x$ the equation $y=\frac{x}{1-x^2}\Leftrightarrow yx^2+x-y=0$ with the restriction $-1<x<1$. When $y=0$, we get $x=0$. When $y\ne 0$, this is a quadratic equation with roots $$x=\frac{-1\pm\sqrt{1+4y^2}}{2y}.$$
Now you have to examine the two roots and determine whet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4425640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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maximum of reciprocals of integers Suppose you have n integers, such that $\sum_{i=1}^n \frac{1}{a_i}<1$. What is the maximum value of this sum?
The answers for $n=1,2,3,4$ are $1-\frac{1}{2}, 1-\frac{1}{6}, 1-\frac{1}{42}, 1-\frac{1}{42*43}$(checked by hands).
Could it be anyhow proven that this sequence is exactly co... | Claim:
$\max \{ \sum_{i=1}^n \frac{1}{a_i} \} = 1- \frac{1}{A_n}$ , where $A_n=(A_{n-1})^2+A_{n-1}$ with $A_0=1$.
Proof:
Let the claim be true for some $n=k$. That is,
$\max \{ \sum_{i=1}^k \frac{1}{a_i} \} = 1- \frac{1}{A_k}$ .
Now we need to find $$\max \{ \sum_{i=1}^{k+1} \frac{1}{a_i} \} $$.
We already know
$\max \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4426549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Trigonometric integral inequality Let us consider the integral equation
\begin{equation}
f(x)=\lambda \int_{0}^{\pi} \cos (x-y) f(y) \hspace{1mm}dy+g(x), \quad x \in[0, \pi],
\end{equation}
where $f$ is an unknown function on $[0, \pi]$, $g(x)$ is a given continuous function on $[0, \pi]$ and $\lambda$ is a given r... | Since
$$
\cos (x-y) =\cos x \cos y + \sin x \sin y,
$$
we have that the equation in question is equivalent to
$$
f(x)=\lambda \cos x \int_0^\pi f(y)\cos y \hspace{1mm}dy + \lambda \sin x \int_0^\pi f(y)\sin y \hspace{1mm}dy + g(x).
$$
We see that $f$ is the function of the form
\begin{equation}
f (x) =a\cos x +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4432972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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How to find $\arg(z)$ and $|z|$? How to find $|z|$ and $\arg (z)$
$z$ is complex number and
$z$ is defined by $$z=\left(\cos\frac\pi5+i\sin\frac\pi5\right)^{15}\cdot(3-3i)^{20}$$
I`ve tried to behave it like $$e^{i15\frac\pi5}\cdot e^{i20\frac\pi4}$$ and got in result = $$e^{i3\pi}\cdot e^{i5\pi}=e^{i8\pi}$$ which give... | Just remember some basic properies of $\arg$ and absolue value: If a value is given in polar form, then the polar coordinates are obtained. With $r, \varphi\in \Bbb R$ and $r\geqslant 0$:
$$\begin{align*}
|r\cdot e^{i\varphi}| &= r \\
\arg (r\cdot e^{i\varphi}) &\equiv \varphi
\end{align*}$$
Where $\equiv$ means that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4436066",
"timestamp": "2023-03-29T00:00:00",
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Find minimum of $\sqrt{a}+\sqrt{b}+\sqrt{c}$, given that $a,b,c \ge 0$ and $ab+bc+ca+abc=4$
Find the minimum of $\sqrt{a}+\sqrt{b}+\sqrt{c}$, given that $a,b,c \ge 0$ and $ab+bc+ca+abc=4$.
I can prove that $\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}=1,$ because:
$$ab+bc+ca+abc=4 \\ \implies (a+2)(b+2)(c+2)=(a+2)(b+2)+... | Here is a sketch of the solution (you need to fill in the details).
One might use the following substitution
$$
a=\frac{2x}{y+z},~b=\frac{2y}{z+x},~c=\frac{2z}{x+y},
$$
where $a,b,c>0$ (why do they exist?). Thus, it remains to find the minimum (or rather infimum) of
$$
F(x,y,z)=\sqrt{\frac{2x}{y+z}}+\sqrt{\frac{2y}{z+x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4436752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
A pattern that leads to regular continued fractions of quadratic irrationals The following expression can be obtained by converting the continued fraction of quadratic irrationals to single fraction.
$$
\sqrt{N} = \frac{b\sqrt{N}+aN}{a\sqrt{N}+b}
$$
The equation holds for any values of $a$ and $b$. However, by calculat... | your matrix is the generator of the (oriented) automorphism group of the quadratic form $x^2 - N y^2.$ Given the smallest positive (nonzero) $u,v$ solving the Pell equation $u^2 - N v^2 = 1,$ your matrix is
$$
A =
\left(
\begin{array}{cc}
u & Nv \\
v & u
\end{array}
\right)
$$
and your fraction is
$$ \frac{u ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4443849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find all integers $n$ for which $x^6 + nx^2 − 1$ can be written as a product of two non-constant polynomials with integer coefficients. Find all integers $n$ for which $x^6 + nx^2 − 1$ can be written as a product of two non-constant polynomials with integer coefficients.
I first tried expanding: $(x^3+ax^2+bx+1)(x^3+cx... | This is not too difficult to do case-by-case. Write $f(x)=x^6+nx^2-1$.
*
*By the rational root theorem, the only possible linear factors are $x\pm1$. They are both factors if and only if $n=0$, when $f(x)=(x-1)(x+1)(x^4+x^2+1)$.
*The OP is well on their way to solving the case of two cubics. As the other steps also ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4445291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Simplify $\frac{1+\sin\alpha-2\sin^2\left(45^\circ-\frac{\alpha}{2}\right)}{4\cos\frac{\alpha}{2}}$ Simplify $$\dfrac{1+\sin\alpha-2\sin^2\left(45^\circ-\dfrac{\alpha}{2}\right)}{4\cos\dfrac{\alpha}{2}}$$
I am reading the solution of the authors and I really don't see how $$\dfrac{1+\sin\alpha-2\sin^2\left(45^\circ-\df... | Only one thing changes, namely: $2\sin(45^\circ - a/2)^2 \rightarrow 1-\cos(90^\circ - a).$
So recall the double angle formula: $\cos(2\theta) = 1 - 2\sin(\theta)^2$
Then by rearrangement $2\sin(\theta)^2 = 1 - \cos(2\theta)$.
Let $\theta = 45^\circ - a/2$, then $2\sin(45^\circ - a/2)^2 = 1-\cos(2\cdot (45^\circ - a/2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4445806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find the range of $\frac{\sqrt{(x-1)(x+3)}}{x+2}$
Find the range of $\frac{\sqrt{(x-1)(x+3)}}{x+2}$
My Attempt:$$y^2=\frac{x^2+2x-3}{x^2+4x+4}=z\\\implies x^2(z-1)+x(4z-2)+4z+3=0$$
Discriminant greater than equal to zero, so, $$(4z-2)^2-4(z-1)(4z+3)\ge0\\\implies z\le\frac43\\\implies -\frac2{\sqrt3}\le y\le\frac2{\... | The problem is that squaring an equation may introduce extraneous solutions. You either have to check each solution (by substituting it into the original equation), or you use equivalent equations throughout the solution process which ensures extraneous solutions won't occur. Taking the second approach, your equation $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4447519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
The characteristic polynomial of this family of matrices I'm looking at the following family of $n\times n$ matrices. The entries are 0 everywhere except above and below the diagonal. Above it takes values from $1 \to n-1$ and below from $ -n +1 \to -1$. Example when $n=4$:
$\left[
\begin{matrix}
0 & 1 & 0 & 0 \\
-3 & ... | Let $D=\operatorname{diag}(1,i,i^2,\ldots,i^{n-1})=\operatorname{diag}(1,i,-1,-i,1,i,-1,-i,\ldots)$. Then $D^{-1}AD=iK$ where
$$
K=\pmatrix{
0&1\\
n-1&0&2\\
&n-2&\ddots&\ddots\\
& &\ddots&\ddots&\ddots\\
& & &\ddots&0 &n-1\\
& & & & 1 &0}.
$$
is known as the Kac matrix. The spectrum of $K$ is given by... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4452105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Showing $\int _{0} ^{\pi/4} \frac{\cos^{2022}(x)}{\sin^{2022}(x) + \cos^{2022}(x) } dx \approx \frac{\pi}{4}$
Show that $$\int_{0} ^{\pi/4} \frac{\cos^{2022}(x)}{\sin^{2022}(x) + \cos^{2022}(x) } dx \approx \frac{\pi}{4}$$
My method was this: I tried using $x \to \pi/4-x$ conversion but that doesn't lead to common de... | Let
$$I_n := \int_{0} ^{\pi/4} \frac{\cos^n(x)}{\sin^n(x) + \cos^n(x) } \,\mathrm{d}x.$$
We have
$$I_n = \frac{\pi}{4} - \int_{0} ^{\pi/4} \frac{\sin^n(x)}{\sin^n(x) + \cos^n(x) } \,\mathrm{d} x .$$
Clearly, $I_n < \frac{\pi}{4}$.
Also, we have
\begin{align*}
I_n &> \frac{\pi}{4} - \int_{0} ^{\pi/4} \frac{\sin^n(x)}{ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4453857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to transform triple integral $\iiint_\Omega \sqrt{1- \frac{x^2}{a^2}- \frac{y^2}{b^2} - \frac{z^2}{c^2} }\ dx dy dz$ I have stumbled across this triple integral
$$\iiint_\Omega \sqrt{1- \frac{x^2}{a^2}- \frac{y^2}{b^2} - \frac{z^2}{c^2} }\ dx dy dz$$
where
$$\Omega =\left\{(x,y,z)\in{\cal{R}}^3\ \bigg| \ \frac{x^2... | Make the variable changes
$x= a u$, $ y= b v$, $z= c w$, and then integrate in spherical coordinates
\begin{align}
&\iiint_{\frac{x^2}{a^2}+ \frac{y^2}{b^2}
+ \frac{z^2}{c^2}\le1 }\sqrt{1- \frac{x^2}{a^2}- \frac{y^2}{b^2}
- \frac{z^2}{c^2} }\ dx dy dz \\
=&\ abc \iiint_{u^2 +v^2+w^2\le1 }\sqrt{1- {u^2}- {v^2}
- {... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4456988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let p be a prime. Prove that $x^4 + 4 y^4 = p$ has integer solutions if and only if $p = 5$. In this case, find the solutions for x and y. I have tried to factor
$$x^4 + 4 y^4 = p$$
using the Sophie Germain's Identity, as someone suggested in the comments, which yields:
$$((x+y)^2 + y^2)((x-y)^2 + y^2) = p$$
Since p is... | Here's a hint:
$$x^{4} + 4y^{4} = (x^2)^2 + (2y^{2})^{2} + 4x^{2}y^{2} - 4x^{2}y^{2} = (x^{2} + 2y^{2})^{2} - (2xy)^{2}.$$ Can you factor this further?
If $x^{4} + 4y^{4} = p$ is prime, what does this say about the factors of $x^{4} + 4y^{2}$?
(Edited to give more details)
So, we have $$x^{4} + 4y^{4} = (x^{2} + 2y^{2}... | {
"language": "en",
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"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Let $D,m$ be relatively prime integers with $m$ odd. Then $D \equiv 0,1 \pmod 4$ and $D \equiv b^2 \pmod m$ implies that $D \equiv b^2 \pmod{4m}$ This is from a proof in David A. Cox's Primes of the Form $x^2+ny^2$:
Lemma 2.5 Let $D\equiv 0,1 \bmod 4$ be an integer and $m$ be an odd integer relatively prime to $D$. T... | I don't think this holds unless we assume $b$ is odd. Let $D=81$ and $m=45$. Then $(D \pmod {45}) =36=6^2$ but $(D \pmod {4×45})=81=36+49$ $=36+m$.
If on the other hand $D \pmod m$ is an odd square $b^2$, then because both $D$ and $m$ are odd, it follows that the equation $D=b^2+km$ must hold for an integer $k$ that mu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4460235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Finding distance between a point and parametric equations This problem is from the parametric and trigonometric coordinate systems of the Art of Problem Solving Precalculus book:
Find the smallest distance between the point $ (1,2,3) $ and a point on the graph of the parametric equations $ x = 2-t, y=4+t, z=3+2t.$
I tr... | You have the given point $P_0 = (1, 2, 3) $
And the curve (which is a line) is given by
$Q(t) = (2,4,3) + t ( -1, 1 , 2) $
The vector extending from $P_0$ to a point on the line $Q(t) $ is
$V(t) = Q(t) - P_0 = (2, 4, 3) + t (-1, 1, 2) - (1, 2, 3) = (1, 2, 0) + t (-1, 1, 2) $
And you want this vector to be perpendicular... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4465115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Simplifiying the trigonometry equation to show that triangle is isosceles In a triangle $ABC$ , $X$ is a point inside it , its given that Angle $BAX = 10^\circ$ , Angle $ABX = 20^\circ$ , Angle $XAC = 40^\circ$ and Angle $XCA = 30^\circ$, prove that triangle is isosceles.
My method was using trigonometry to arrive at ... | Here is an approach using the Law of Sines and the Law of Cosines, that is not too bad.
$$
\begin{align}
BX
&=AB\,\frac{\sin\left(10^{\large\circ}\right)}{\sin\left(150^{\large\circ}\right)}\tag{1a}\\[6pt]
&=2AB\sin\left(10^{\large\circ}\right)\tag{1b}\\[6pt]
CX
&=AX\,\frac{\sin\left(40^{\large\circ}\right)}{\sin\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4468008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Spivak, Ch. 13 "Integrals", Prob. 2: Show $\int_0^b x^4 dx=\frac{b^5}{5}$ by finding unique number such that $L(f,P)\leq \int_0^b x^4 \leq U(f,P)$? Consider the problem (Spivak, Chapter 13 "Integrals", problem 2) of showing that $\int_0^b x^4 dx=\frac{b^5}{5}$ by finding the unique number $\int_0^b x^4$ such that
$$L(f... | Inequality (1) is algebraically the same as the inequality
$$
15n^3-10n^2+1>0,\tag{1'}
$$
which you can prove by rewriting the LHS in the form:
$$
15n^3-10n^2+1 = 15n^2(n-\frac23)+1
$$ and noting that $n^2(n-\frac23)$ is positive for all integers $n\ge1$.
Inequality (2) is the algebraically the same as
$$
15n^3+10n^2>1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4469191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to solve long integration by partial fraction decomposition problems faster? Some problems are just too time consuming for short exam times what is the fastest way to solve problems like this one for example $$\int \frac{5x^4-21x^3+40x^2-37x+14}{(x-2)(x^2-2x+2)^2}dx$$
| With $x=t+1$
\begin{align}
&\ \frac{5x^4-21x^3+40x^2-37x+14}{(x-2)(x^2-2x+2)^2}
= \frac{5t^4-t^3+7t^2+1}{(t-1)(t^2+1)^2}
= \frac{A}{t-1}+ {P(t)}
\end{align}
where $A = \lim_{t\to 1}\frac{5t^4-t^3+7t^2+1}{(t^2+1)^2}=3
$ and
\begin{align}
P(t)= &\ \frac{5t^4-t^3+7t^2+1}{(t-1)(t^2+1)^2}
-\frac{3}{t-1}\\
= &\ \frac{2t^3+t^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4469257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Integral with Weierstrass substituion, limits gone wrong I'm trying to evaluate the following integral with a Weierstrass substitution:
$$
\int_{\pi/3}^{4\pi/3} \frac{3}{13 + 6\sin x - 5\cos x} \text{d}x
$$
This comes out to about 0.55 when evaluated numerically. I thought it would be possible to rewrite the integral u... | Take care of the discontinuity at $x=\pi$ with the half angle substitution $t=\tan\frac x2$
\begin{align}
&\int_{\pi/3}^{4\pi/3} \frac{3}{13 + 6\sin x - 5\cos x} dx
\overset{t=\tan\frac x2}=\bigg(\int_{\frac1{\sqrt3}}^\infty +\int_{-\infty}^{-\sqrt3}\bigg)
\frac3{4+6t+9t^2}dt
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4471956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is there any method other than Feynman’s Integration Technique to find $ \int_{0}^{\frac{\pi}{2}} \ln \left(a \cos ^{2} x+b \sin ^{2} x+c\right) d x?$ We are going to find the formula, by Feynman’s Integration Technique, for
$$\int_{0}^{\frac{\pi}{2}} \ln \left(a \cos ^{2} x+b \sin ^{2} x+c\right) d x,$$
where $a+c$ $\... | Use the short-hands $p^2=a+c$, $q^2=b+c$ and $r=\frac{p-q}{p+q}$ to rewrite the integral as
\begin{align}
I=&\int_{0}^{\frac{\pi}{2}} \ln \left(a \cos ^{2} x+b \sin ^{2} x+c\right) d x\\
=& \int_{0}^{\frac{\pi}{2}} \ln \left(p^2 \cos ^{2} x+q^2 \sin ^{2} x\right) d x\\
= & \int_{0}^{\frac{\pi}{2}}\bigg(\ln\frac{(p+q)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4472100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Evaluate $\int_{-\infty}^\infty \frac{1}{(x^2 + D^2)^2}dx$. $$\int_{-\infty}^\infty \frac{1}{(x^2 + D^2)^2}dx$$
Edit : $D> 0$.
My work:
Let $x = D\tan \theta$
$$\int_{-\infty}^\infty \frac{1}{(x^2 + D^2)^2}dx=\int_{-\infty}^\infty \frac{1}{(D^2\sec^2 \theta)^2}dx$$
$$=\int_{-\infty}^\infty \frac{1}{(D^2\sec^2 \theta)^2... | I’ll suggest another method, involving Integration By Parts. Let the integral to be found be denoted by J. We start by performing an integration by parts of the function $\frac{1}{x^2+D^2}$:
$$I= \int \frac{1}{x^2+D^2} dx= \frac{x}{x^2+D^2}-\int \left(-\frac{x(2x)}{(x^2+D^2)^2}\right)dx$$ $$= \frac{x}{x^2+D^2}+2 \i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4477389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Distribution of amount of green balls drawn Urn of type A contains 4 green and 3 blue balls. Urn of type B contains 4 blue and 5 green balls.
There are three urns of type A and two urns of type B. We pick one urn at random (out of 5) and we draw 3
balls (with a single draw) out of the chosen urn. Let X denote the numbe... | To better understand the question I think it would be better to consider a random variable $Y$ to the problem. Let $Y$ be the random variable that indicates which urn was chosen. Note that
$$
\begin{array}{ll}
\mathbb{P}(Y=A)=3/5
&
\mathbb{P}(Y=B)=2/5
\\
\mathbb{P}(X=n | Y=A)=\frac{\binom{4}{n}\binom{3}{3-n}}{\binom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4478158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solving the ODE $(x^2 y-1)y'-(xy^2-1)=0$ I need to solve $(x^2 y-1)y'-(x y^2-1)=0$.
Not sure how to approach this ODE, would love some help regarding solving it or any useful resources.
| Too long for a comment
Using @projectilemotion's answer, we can easily solve the cubic equation
$$y^3-3x y^2+\frac{3 (2 C+1) x^2}{2 C}y-\frac {(2 C+1) x^3+2 }{2C}=0$$
For $C=1$, the solution is
$$y=x \left(1-\sqrt{2} \sinh \left(\frac{1}{3} \sinh ^{-1}\left(\sqrt{2}\frac{
\left(x^3-1\right)}{x^3}\right)\right)\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4478931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Convergence of $\sum_{n=1}^\infty \frac{\sum_{i=1}^n\frac{1}{ \sqrt i}}{n^2}$ How can I prove the following sequence converges?
$$\sum_{n=1}^\infty \frac{\sum_{i=1}^n\frac{1}{ \sqrt i}}{n^2}$$
I tried everything.
Could not find any candidates for comparison test, and failed to find an upper bound.
Any hints will be app... | Using generalized harmonic numbers, you face
$$S_p=\sum_{n=1}^p \frac{H_n^{\left(\frac{1}{2}\right)}}{n^2}$$
For large values of $n$
$$H_n^{\left(\frac{1}{2}\right)}=2\sqrt{n}+\zeta \left(\frac{1}{2}\right)+\frac{1}{2 \sqrt{n}}\Bigg[1-\frac{1}{12 n}+\frac{1}{192 n^3}-\frac{1}{512
n^5}+O\left(\frac{1}{n^6}\right) \Bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4479824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Proving $(ac+bd)^2+(ad-bc)^2=(a^2+b^2)(c^2+d^2)$ with various solutions.
$(ac+bd)^2+(ad-bc)^2=(a^2+b^2)(c^2+d^2)$
Solutions in the answers.
$\ \\ \ \\ \ \\ \ \\$
Edit) Since this question is closed, I'll add more contexts for this question.
This identity is called "Brahmagupta-Fibonacci identity", which the comment... | There are two possible factorizations.
Here are both of them.
$\begin{array}\\
(a^2+b^2)(c^2+d^2)
&=a^2c^2+a^2d^2+b^2c^2+b^2d^2\\
&=a^2c^2+b^2d^2+a^2d^2+b^2c^2\\
&=a^2c^2+b^2d^2\pm2a^2b^2c^2d^2+a^2d^2+b^2c^2\mp2a^2b^2c^2d^2\\
&=(ac\pm bd)^2+(ad\mp bc)^2
\qquad\text{The two signs are different}\\
\end{array}
$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4479936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 2
} |
Question about conic sections and determinant Now I have found questions which were on the same topic on this site..but not quite one like this.
Let there be six points $p_1=(x_1,y_1),..,p_6= (x_6,y_6) $
Why is it, that if $$ det\begin{pmatrix}1 &1 &1 & 1& 1 & 1 \\ x_1 & x_2 & x_3 &x_4 & x_5 & x_6 \\
y_1 & y_2 & y_3 &y... | Let us consider the following homogeneous system with the matrix you have, but transposed :
$$\begin{pmatrix}1 &1 &1 & 1& 1 & 1 \\ x_1 & x_2 & x_3 &x_4 & x_5 & x_6 \\
y_1 & y_2 & y_3 &y_4 & y_5 &y_6 \\ x_1^2 & x_2^2 & x_3^2 &x_4^2 & x_5^2 & x_6^2 \\ x_1y_1 & x_2y_2 & x_3y_3 & x_4y_4 & x_5y_5 & x_6y_6 \\ y_1^2 & y_2^2 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4482421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Limit of sequence given by $x_{n} = \frac{x_{n-2}+x_{n-3}}{2}$? Let $x_1,x_2,x_3\in\mathbb{R}$ be three distinct real numbers. I am interested in the convergence of the sequence
$$
x_{n} = \frac{x_{n-2}+x_{n-3}}{2},\quad n = 4,5,\ldots
$$
ie,
$$
x_{4} = \frac{x_{2}+x_{1}}{2},\quad x_{5} = \frac{x_{3}+x_{2}}{2},\quad x_... | For the linear recurrence $$x_{n} = \frac{x_{n-2}+x_{n-3}}{2}$$ the characteristic polynomial is
$$r^3=\frac {r+1}2 \implies r_1=1 \qquad r_2=-\frac {1+i} 2\qquad r_3=-\frac {1-i} 2$$
and, as usual, $$x_n=c_1 + c_2 r_1^n+c_3 r_2^n$$
Using $x_1=a$, $x_2=b$, $x_3=c$, this would give
$$x_n=A_n \,a+B_n\, b+C_n\, c$$ with
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4485083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
A cryptogram Diophantine $\overline{ABCDE}=4 \ \overline{AB}^2+\overline{CDE}^2$ ABCDE is a 5-digit number. AB and CDE are 2-digit and 3-digit numbers respectively.
$\overline{ABCDE}=4 \ \overline{AB}^2+\overline{CDE}^2$
Find all possible values of $\overline{ABCDE}$.
Honestly, I designed this problem just for fun, and... | We convert the given equation to the following form.
$$1000(AB)+(CDE)=4(AB)^2+(CDE)^2. $$
Now consider the following quadratic equation.
$$(CDE)^2-(CDE)+4(AB)^2-1000(AB)=0$$
The two roots of this equation are,
$$(CDE)=\dfrac{1\pm \sqrt{1-16(AB)^2+4000(AB)}}{2}.$$
Since $(AB)$ is a 2-digit number, $10\le (AB)\le 99$. Fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4485412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Solving the equation $x^2+(\frac{x}{x+1})^2=\frac54$
Solve the equation$$x^2+\left(\frac{x}{x+1}\right)^2=\frac54$$
I noticed that for $0< x$, both $x^2$ and $\left(\dfrac x{x+1}\right)^2$ are increasing functions so their sum is also increasing and it has only one root which is $x=1$ (by inspection). But I'm not sur... | Simplify as
$$4x^2[(x+1)^2+1]=5(x+1)^2 \implies 4x^4+8x^3+3x^2-10x-5=0$$
Denoting LHS as $f(x)$, note that $f(1)=0 ~\& ~f(-1/2)=0.$ This means that by remainder theorem $$f(x) ~\text{is divisible by}~ (2x+1)(x-1)=2x^2-x-1=g(x)$$
dividing $f(x)$ by $g(x)$ we see that $$f(x)=(2x^2-x-1)(2x^2+5x+5)$$
By solving $2x^2+5x+5=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4493459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Correct version of an inequality Does any one knows the correct version of the inequality:
Let $a_1,\cdots,a_n\in \mathbb R$ and $b_1,\cdots,b_n\in \mathbb R$ with $\sum_{j=1}^n b_j=0$. Then
$$
\left(\sum_{j=1}^n a_jb_j \right)^2\leq \left(\sum_{j=1}^n a_j^2-\left(\sum_{j=1}^n a_j\right)^2\right)\sum_{j=1}^n b_j^2\,,
... | I believe the issue is that rather than sums, those should be means. That is,
$$
\left[\frac{1}{n}\sum_{j=1}^na_j^2 - \left(\frac{1}{n}\sum_{j=1}^na_j\right)^2\right]\left(\frac{1}{n}\sum_{j=1}^n b_j^2\right)\ge\left(\frac{1}{n}\sum_{j=1}^na_jb_j\right)^2
$$
This can also be written more compactly in vector notation:
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4499830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$x^{10}+x^{11}+\dots+x^{20}$ divided by $x^3+x$. Remainder? Question:
If $x^{10}+x^{11}+\dots+x^{20}$ is divided by $x^3+x$, then what is the remainder?
Options: (A) $x\qquad\quad$ (B)$-x\qquad\quad$ (C)$x^2\qquad\quad$ (D)$-x^2$
In these types of questions generally I follow the following approach:
Since divisor is... | $\overbrace{\!\!\bmod \color{c00}{x^2\!+\!1}}^{\color{#c00}{\large x^2\ \equiv\ -1}\!\!}\!:\ f\color{#0af}{\!+\!x^9} =\! (x^9\!+\!x^{13}\!+\!x^{17})\!\!\overbrace{(1\!+\!x\!+\!x^2\!+\!x^3)}^{\large ((\color{#c00}{x^2})^2-1)/(x-1)=\color{#0a0}0}\!\!\equiv\color{#0a0}0\,$ so $\,f \equiv \color{#0af}{-x^9} = -x(\color{#c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4502967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 5
} |
Find $q,r$ if $q^2+r=2000$ Let $a,b \in \mathbb{N}$ such that when $a^2+b^2$ is divided by $a+b$, the quotient is $q$ and remainder is $r$ such that $q^2+r=2000$. Find $q,r$
My try:
Obviously $a \ne b$ for if we have $r=0$ $\implies$ $q=\sqrt{2000} \notin \mathbb{N}$
Without loss of generality, let $a>b$ and $a-b=c$
Th... | From
$$a^2+b^2=q(a+b)+r\ ,\qquad q^2+r=2000$$
we get
$$q^2-(a+b)q+(a^2+b^2-2000)=0\ .$$
For this to have real solutions we need
$$(a+b)^2\ge 4(a^2+b^2-2000)$$
which simplifies to
$$8000\ge 3a^2-2ab+3b^2=2a^2+2b^2+(a-b)^2\ .$$
Therefore $8000>2a^2,\,2b^2$ and so
$$a,b\le63\ .$$
However the largest square below $2000$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4503398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating $ \frac{24}{\pi}\int_0^\sqrt 2\frac{2-x^2}{(2+x^2)\sqrt{4+x^4}}\,\mathrm dx$ I recently came across a problem on definite integration and couldn't solve it despite my efforts. It goes as
$$
\frac{24}{\pi}\int_0^\sqrt 2\frac{2-x^2}{(2+x^2)\sqrt{4+x^4}}\,\mathrm dx
$$
The $2-x^2$ term and the upper limit along... | Integrate as follows
\begin{align}&\ \frac{24}{\pi}\int_0^\sqrt 2\frac{2-x^2}{(2+x^2)\sqrt{4+x^4}}\,dx\\
=&\ \frac{24}{\pi}\int_0^\sqrt2\frac{\frac{2-x^2}{(2+x^2)^2}}{\sqrt{1-\frac{4x^2}{(2+x^2)^2}}}\,
\overset{y=\frac{2x}{2+x^2}}{dx}
=\frac{12}{\pi}\int_0^{\frac1{\sqrt2}}\frac{dy}{\sqrt{1-y^2}}
= \frac{12}{\pi}\cdot\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4503603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Determine the power series representation of the function $f(x)=\sqrt{(4+x)^3}$ and indicate the radius of convergence I want to find a representation of the function mentioned above, so I took into account that:
$$f(x)=\sqrt{(4+x)^3}=8\left(1+\frac{x}{4}\right)^{\frac{3}{2}}$$ and developing the binomial series for $\... | According to the binomial series expansion we obtain
\begin{align*}
\left(1+\frac{x}{4}\right)^{\frac{3}{2}}&=1+\frac{3}{2}\left(\frac{x}{4}\right)+\frac{\frac{3}{4}}{2!}\left(\frac{x}{4}\right)^2+\frac{\frac{-3}{8}}{3!}\left(\frac{x}{4}\right)^3+\cdots\\
&=1+\binom{3/2}{1}\frac{x}{4}+\binom{3/2}{2}\left(\frac{x}{4}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4503835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What's the shortest path to simplify $\tan{85°} \tan{35°} \tan{15°}+\tan{55°} \tan{5°} \tan{75°}+2\cot{50°}$? We have this expression:
$$\tan{85°} \cdot \tan{35°} \cdot \tan{15°}+\tan{55°} \cdot \tan{5°} \cdot \tan{75°}+2\cot{50°}$$
I want this expression to be in its simplest form.
I fed this into the Symbolab calcula... | Consider the identity $\tan(x)\cdot\tan(60°-x)\cdot\tan(60°+x)=\tan(3x)$. We now have
$$\tan5°\cdot\tan55°\cdot\tan65°=\tan15°\implies\tan5°\cdot\tan55°\cdot\tan75°=\cot65°$$
and
$$\tan25°\cdot\tan35°\cdot\tan85°=\tan75°\implies\tan15°\cdot\tan35°\cdot\tan85°=\cot25°$$
The expression now is simplified into $$\tan25°+\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4506254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Sum of all real roots of $\frac{{3{x^2} - 9x + 17}}{{{x^2} + 3x + 10}} = \frac{{5{x^2} - 7x + 19}}{{3{x^2} + 5x + 12}}$?
The sum of real roots of $\dfrac{{3{x^2} - 9x + 17}}{{{x^2} + 3x + 10}} = \dfrac{{5{x^2} - 7x + 19}}{{3{x^2} + 5x + 12}}$ is____
How do I proceed with this type of problem.
Cross multiplying and s... | Suggestion (Hint):
Use the well-known rule:
$$\frac ab=\frac cd \iff \frac {a-b}{b}=\frac {c-d}{d}$$
Then observe that: $$a-b=c-d$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4506668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What periodic function does $\frac{\Gamma(x+1)}{2^x \Gamma(\frac{x}{2}+1)^2}$ tend to? If one graphs $$ y= \frac{\Gamma(x+1)}{2^x \Gamma(\frac{x}{2}+1)^2} = \frac{x!}{2^x \left( \left( \frac{x}{2}\right)!\right)^2} $$
One immediately notices that as $x \rightarrow -\infty$ the function tends to some periodic function ... | For negative values of $x$, $$y= \frac{\Gamma(x+1)}{2^x\, \Gamma(\frac{x}{2}+1)^2} $$ is closer and closer to
$$\sin ^2\left(\frac{\pi x}{2}\right) \csc (\pi (x+1))=-\frac{1}{2} \tan \left(\frac{\pi x}{2}\right)$$ This comes from Euler's reflection formula of the gamma function.
Edit
As @GregMartin commented, I for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4506991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
How many methods are there to find $\int_{0}^{\pi} \frac{d \theta}{a+b \cos \theta}, \textrm{ where }a>b>0$ When I deal with this simple integral, I found there are several methods. Now I share one of them.
Letting $x\mapsto \pi-x $ yields $$I=\int_{0}^{\pi} \frac{d \theta}{a+b \cos \theta}=\int_{0}^{\pi} \frac{d \thet... | Here's an interesting geometric approach I read in one of Jack D' Aurizio's answers (and have never forgotten). It's not as efficient as using a Weierstrass substitution but I believe the geometric interpretation itself merits some sort of recognition.
For a closed and continuous curve $r(\theta)$ where $0\leq\theta\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4508099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 0
} |
Interesting integral $\int_{0}^{\frac{\pi}{4}} \frac{x^{2}}{(x \sin x+\cos x)^{2}} d x$? Noting that
$\displaystyle d(x \sin x+\cos x)=x \cos xdx,\tag*{} $
we have
$\displaystyle \int_{0}^{\frac{\pi}{4}} \frac{x^{2}}{(x \sin x+\cos x)^{2}} d x=-\int_{0}^{\frac{\pi}{4}} \frac{x}{\cos x} d\left(\frac{1}{x \sin x+\cos x}\... | \begin{align}
&\int_{0}^{\frac{\pi}{4}} \frac{x^{2}}{(x \sin x+\cos x)^{2}} d x\\
=& \int_{0}^{\frac{\pi}{4}} \frac{x^{2}\sec^2x}{(1+x \tan x)^{2}} d x
= \int_{0}^{\frac{\pi}{4}}
\frac{1}{(1+x \tan x)^{2}}+ \frac{x^{2}\sec^2x -1}{(1+x \tan x)^{2}} \ d x\\
=& \ \left( \frac1{x+\cot x} - \frac{x}{1+x\tan x}\right)_0^{\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4508201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Find the limit of $\frac{1+\frac13+\frac{1}{3^2}+...+\frac{1}{3^n}}{1+\frac15+\frac{1}{5^2}+...+\frac{1}{5^n}}$ Find the limit $$\lim_{n\to\infty}\dfrac{1+\frac13+\frac{1}{3^2}+...+\frac{1}{3^n}}{1+\frac15+\frac{1}{5^2}+...+\frac{1}{5^n}}$$
For the numerator we have $1+\frac13+\frac{1}{3^2}+...+\frac{1}{3^n}=\dfrac{1-\... | You were very close:
$$\lim_{n\to\infty}\dfrac{1+\frac13+\frac{1}{3^2}+...+\frac{1}{3^n}}{1+\frac15+\frac{1}{5^2}+...+\frac{1}{5^n}}=\lim_{n\to\infty}{\dfrac{\frac32-\frac12.\frac{1}{3^n}}{\frac54-\frac14.\frac{1}{5^n}}}$$
as you correctly derived.
If you simply use $\lim_{n\to\infty}\frac{1}{3^n}=\lim_{n\to\infty}\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4509654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Solve the differential equation: $6 \cos^2(x) \dfrac{dy}{dx} -y \sin(x)+2y^4 \sin^3(x)=0$ I have the following differential equation before me:
$6 \cos^2(x) \dfrac{dy}{dx} -y \sin(x)+2y^4 \sin^3(x)=0$
I tried solving it by reducing to Bernoulli form of first order differential equation.
I divided both sides of the equa... | As @abcdefu already did, after $y=\frac{1}{\sqrt[3]{u}}$, we have
$$2 \cos ^2(x)\, u'+ \sin (x)\, u=2 \sin ^3(x)$$
The solution of the homegeneous is simple
$$u=C\,e^{-\frac{\sec (x)}{2}}$$ Variation of parameters leads to
$$\cos ^2(x) e^{-\frac{\sec (x)}{2}} C'=\sin ^3(x) \implies C'=\sin (x) \tan ^2(x) \,e^{\frac{\se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4510057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
If $x^2+2(\alpha-1)x-\alpha+7=0$ has distinct negative solutions... Let $\displaystyle{ \alpha }$ be real such that the equation $\displaystyle{ x^2+2(\alpha-1)x-\alpha+7=0 }$ has two different real negative solutions. Then
*
*$ \ \displaystyle{ \alpha<-2 }$ ;
*$ \ \displaystyle{ 3<\alpha<7 }$ ;
*it is impossible... | The quadratic polynomial $ \ x^2 + 2(\alpha-1)x - \alpha+7 \ \ $ in "vertex form" is $$ \ ( \ x \ + \ [\alpha - 1] \ )^2 \ + \ (7 - \alpha) - (\alpha - 1)^2 \ \ = \ \ ( \ x \ + \ [\alpha - 1] \ )^2 \ - \ ( \ \alpha^2 \ - \ \alpha \ - \ 6 \ ) $$
$$ = \ \ ( \ x \ + \ [\alpha - 1] \ )^2 \ - \ ( \ \alpha \ + \ 2 \ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4511125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Prove $\cot\frac{\theta}{2} - 2\cot\theta = \tan\frac{\theta}{2}$ This is a math exercise I am doing for my year 10 math class. Basically I would like to show how I would prove that $\cot\frac{\theta}{2} - 2\cot\theta = \tan\frac{\theta}{2}$
I was able to expand the LHS and was able to use fractions to get that $$\cot\... | $\begin{align}\frac{1}{2}(\cot\frac{\theta}{2} - \tan\frac{\theta}{2})&=\frac{1}{2}\frac{1-\tan^2\frac{\theta}{2}}{\tan\frac{\theta}{2}}\\& =\frac{1}{\frac{2\tan\frac{\theta}{2}}{1-\tan^2\frac{\theta}{2}}}\\&=\frac{1}{\tan\theta} \\&=\cot\theta\end{align}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4511842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Show that $\cos^2\left(x\right)\left(2+\cos 2x\right)-2 \cos^2\left(x+\frac{\sin(2x)}{4}\right)\ge0$ Show that for all $x\in\mathbb{R},$
\begin{equation}
\cos^2\left(x\right)\left(2+\cos 2x\right)-2 \cos^2\left(x+\frac{\sin 2x}{4}\right)\ge0
\end{equation}
I have checked this numerically and this is correct but I canno... | If you expand the first term on the LHS, and add the second term on the LHS to both sides, you'll get
$2\cos^2(x) + \cos^2(x)\cos(2x) \geq 2\cos^2(x+\frac{\sin(2x)}{4})$
Now, using the identity $\cos(2x)+1 = 2\cos^2(x)$, you get:
$\cos(2x)+1 + \frac{\cos(2x)+1}{2}\cos(2x) \geq \cos(2x+\frac{\sin(2x)}{2})+1$
Subtract $1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4514479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Proof of Implicit Differentiation (showing a statement is true)
Prove that if $x^2+y^2-2y\sqrt{1+x^2} = 0$, then $dy/dx = x/\sqrt{1+x^2}$.
Whilst I have implicitly differentiated in terms of x in order to derive that
$$dy/dx = (-x+2xy/\sqrt{1+x^2})/(y\sqrt{1+x^2}-1-x^2)$$
however I am unsure as to what my next steps ... | We find from the given relation $ \ x^2 + y^2 \ = \ 2y\sqrt{1+x^2} \ $ that
$$ 2x \ + \ 2y·\frac{dy}{dx} \ \ = \ \ 2·\frac{dy}{dx}·\sqrt{1+x^2} \ + \ 2y·\frac12·\frac{1}{\sqrt{1+x^2}}·2x \ $$
$$ \Rightarrow \ \ ( \ y \ - \ \sqrt{1+x^2} \ ) ·\frac{dy}{dx} \ \ = \ \ y· \frac{1}{\sqrt{1+x^2}}· x \ - \ x \ \ \Ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4515483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Suppose the number n = 90k is the smallest integer that has exactly ninety divisors (all positive integers) which includes 1 and n. Find k Suppose the number n = 90k is the smallest integer that has exactly ninety divisors (all
positive integers) which includes 1 and n. Find k.
I know this question has been answered, b... | We have, $N=90k=2\cdot3^2\cdot5\cdot(k)$
Number of divisors of a number is product of the $(\text{powers}+1)$ of each of the it's prime factors.
Since we want exactly $90$ divisors, so the product would look like:
$$\begin {align} 90 &=2\cdot3^2\cdot5 \\ &= 2\cdot3\cdot3\cdot5 \tag A \\ &= 6\cdot3\cdot5 \tag B \\ &= 2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4516322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
The sum of $n$ terms in $1 \cdot 2+2 \cdot 3+3 \cdot 4+\ldots$ I am just confused, considering we can take $1 \cdot 2$ as the first term then we get the $n$th term as $n(n+1)$ so the sum of $n$ terms would be $\frac{n(n+1)(2n+1)}{6}$ + $\frac{n(n+1)}{2}$ but let's assume $0 \cdot 1$ as the first term then the $n$th ter... | $S_1=1⋅2+2⋅3+3⋅4+\dots$
$T_n=n(n+1)$
$S_1=\frac{n(n+1)(2n+1)}{6}\bf+\frac{n(n+1)}{2}$
$S_2=1⋅2+2⋅3+3⋅4+\dots$
$T_n=(n-1)n$
$S_2=\frac{n(n+1)(2n+1)}{6}\bf-\frac{n(n+1)}{2}$
As the bold part indicates, the summation isn't the same.
Let's put $\bf n=1$:
$S_1=1⋅2=2$
$S_1=\frac{1(2)(3)}{6}\bf+\frac{1(2)}{2}=1+1=2$
$S_2=0⋅1=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4519061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
I can't come up with the intended solution to $\frac{a^2+b^2}{2a^5b^5} + \frac{81a^2b^2}{4} + 9ab>18$ I was going trough some easy algebra problems when I encountered
$$
\frac{a^2+b^2}{2a^5b^5} + \frac{81a^2b^2}{4} + 9ab>18.
$$
As you can see the problem is easily solvable with AM > GM
I fairly quickly came up with thi... | Let functions $A$ and $G$ denote the arithmetic and geometric means of their arguments. Then:
$$x := \frac{a^2+b^2}{2a^5b^5} + \frac{81a^2b^2}{4} + 9ab$$
$$= A(\frac{a^2}{a^5b^5}, \frac{b^2}{a^5b^5}) + \frac{81a^2b^2}{4} + 9ab$$
$$= A(\frac{1}{a^3b^5}, \frac{1}{a^5b^3}) + \frac{81a^2b^2}{4} + 9ab$$
$$\ge G(\frac{1}{a^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4519399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Arc length of curve stuck with integration Data from exercise
$$y=\frac{4}{3}x^2+2\\
x\in[-1,1]$$
Formula for length of curve
$$L=\int_a^{b}\sqrt{1+(f(x)')^2}\ dx$$
So far i have
$$y'=\frac{8}{3}x$$
$$\int_{-1}^{1}\sqrt{1+\frac{64}{9}x^2}\ dx$$
Substition
$$t^2=\frac{64}{9}x^2$$
$$t=\frac{8}{3}x$$
$$\frac{3}{8}dt=dx$$
... | Let $x = \sinh \theta \implies$
$$
\begin{align}
I &= \int \sqrt{1+x^2}dx \\
&= \int \sqrt{1+\sinh^2 \theta}\cosh \theta d \theta \\
&= \int \cosh^2 \theta d \theta \\
&= \frac{1}{2}\int 1 + \cosh 2 \theta d \theta \\
&= \frac{1}{2}\theta+\frac{1}{4}\sinh 2 \theta + c\\
&= \frac{1}{2}\sinh^{-1}x+\frac{1}{2}x\sqrt{1+x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4519841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Solving $\cos3x=-\sin3x$, for $x \in [0,2\pi]$ For my first step, I would choose to either divide both sides or add sin3x to both sides. My professor in Pre-Calc told me I shouldn't divide any trig functions from either side. Despite this, I can only find a solution this way.
$$\frac{\cos3x}{-\sin3x}=1$$
$$-\tan3x=1$$
... | though ur answer is not wrong, as long as u show that $\sin(3x) \neq 0$
a better way to solve would be
$\cos( 3x) + \sin(3x) = 0$
so $\sqrt{2}\sin(3x + \frac{\pi}{4}) = 0$
let y= $3x + \frac{\pi}{4}$
$ y \in [\frac{\pi}{4}, 6\pi+\frac{\pi}{4}]$
$\sin(y) =0$
so $ y \in \{\pi , 2\pi , 3\pi , 4\pi , 5\pi , 6\pi\} $
so $x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4521724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Solve for all $x,y\in N$ and $x^3-3^y = 2015$ Question:
Solve for all $x,y\in N$ and $x^3-3^y = 2015$
So far I have tried $\mod(7)$ and $\mod(8)$ but I am not sure if I am in the right path or not I will be happy if someone can toss me a hint.
| Use Metin's idea of "reject solution of too large" and
someone's idea of mod $13$ -> $3\mid m$, I reach a solution.
Consider mod $13$, then
$3^y\equiv1,3,9 \pmod {13}$
$x^3\equiv1,5,8,12 \pmod {13}$
Thus, $3^y\equiv1 \pmod {13}, x^3\equiv1 \pmod {13}, 3\mid y$
Let $y=3z, k=3^z$, so $x>k>0$
Then $x^3-3^y = x^3-(3^z)^3 =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4522074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is there a hypergeometric function form of the quadratic formula that converges for all a,b,c? After reading this question about a general formula for roots, I was curious about the simple case of $N=2$.
As is well known to school children, if $ ax^2 + bx + c = 0 $, then the roots are $ \frac{-b \pm \sqrt{b^2-4ac}}{2a}... | $$S_n=-\frac{c}{b} \sum_{k=0}^{n} \binom{-\frac{1}{2}+k}{k} \left( \frac{4ac}{b^2} \right)^k \frac{1}{(k+1)} $$
$$S_n=-\frac{b \left(1-\sqrt{1-\frac{4 a c}{b^2}}\right)}{2 a}+\frac{c }{b
}\,\frac{\binom{n+\frac{1}{2}}{n+1}}{n+2 }\,\left(\frac{4a c}{b^2}\right)^{n+1}\, _2F_1\left(1,\frac{2n+3}{2};n+3;\frac{4 a c}{b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4523224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How to proceed with a definite integral of this form? How do I find the definite integral of this sort of function?
$$ \int_m^n \frac{\sin(a \sin(\sqrt{(b \lfloor x\rfloor^2 + b d + 1)}))}{a \sin(\sqrt{(b \lfloor x\rfloor^2 + b d + 1)})} dx $$
This function is actually a form of the $ \operatorname{sinc} $ functi... | \begin{eqnarray}
\int_m^n \frac{\sin(a \sin(\sqrt{(b \lfloor x\rfloor^2 + b d + 1)}))}{a \sin(\sqrt{(b \lfloor x\rfloor^2 + b d + 1)})} dx&=&\sum_{k=m}^n\int_k^{k+1} \frac{\sin(a \sin(\sqrt{(b k^2 + b d + 1)}))}{a \sin(\sqrt{(b k^2 + b d + 1)})} dx\\
&=&\sum_{k=m}^n \frac{\sin(a \sin(\sqrt{(b k^2 + b d + 1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4523817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Sum of $\sum_0^\infty \frac1{n^4+a^4}$ How to find the following series as pretty closed form by $a$?
$$S=\sum_0^\infty \frac1{n^4+a^4}$$
I first considered applying Herglotz trick, simply because the expressions are similar. So I changed it like this...
$$2S-a^{-4}=\sum_{-\infty}^\infty \frac1{n^4+a^4}$$
However, the ... | We start with partial fractions,
$$\frac{1}{n^4+a^4} = \frac{i}{2a^2}\left(\frac{1}{n^2 + i a^2} - \frac{1}{n^2 - i a^2}\right) \tag 1$$
And then note that (Series expansion of $\coth x$ using the Fourier transform)
$$\coth x = \frac{1}{x} + 2 \sum_{n=1}^\infty \frac{x}{x^2+\pi^2n^2}$$
which we rearrange to give:
$$\su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4524326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Find the all possible values of $a$, such that $4x^2-2ax+a^2-5a+4>0$ holds $\forall x\in (0,2)$
Problem: Find the all possible values of $a$, such that
$$4x^2-2ax+a^2-5a+4>0$$
holds $\forall x\in (0,2)$.
My work:
First, I rewrote the given inequality as follows:
$$
\begin{aligned}f(x)&=\left(2x-\frac a2\right)^2+\fr... | I want to bring the most straightforward way to calculate this problem systematically. Also, you can check the attached graph for better representation. I write the quadratic equation $f_x(a)=4x^2-2ax+a^2-5a+4$ w.r.t. $a$ as an unknown parameter. We have:
$f_x(a)=a^2-(2x+5)a+(4x^2+4)>0$.
So, $a$ must be greater than th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4528746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 11,
"answer_id": 9
} |
How to solve this group of equations? $$\dfrac{yz(y+z-x)}{x+y+z}=p \tag1$$
$$\dfrac{xz(x+z-y)}{x+y+z}=q \tag2$$
$$\dfrac{xy(x+y-z)}{x+y+z}=r \tag3$$
This group of equations comes from the problem "if the distances of incenter of an triangle to its vertices are $\sqrt{p}$, $\sqrt{q}$ and $\sqrt{r}$. Find the length of s... | Rearranging, we get
$$p+\frac{2xyz}{x+y+z}=yz$$
$$q+\frac{2xyz}{x+y+z}=xz$$
$$r+\frac{2xyz}{x+y+z}=xy$$
if we multiply two of these equations and divide by the third,
$$\implies x=\sqrt{\frac{(q+a)(r+a)}{(p+a)}}$$
$$\implies y=\sqrt{\frac{(r+a)(p+a)}{(q+a)}}$$
$$\implies z=\sqrt{\frac{(p+a)(q+a)}{(r+a)}}$$
where $a=2xy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4530453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
$P(x)=x^2+ax+b$ and $Q(x)=x^2+a_1x+b_1$ and $P(x)\cdot Q(x)$ has at most one real root then prove that the claims given below are true.
If $P(x)=x^2+ax+b$ and $Q(x)=x^2+a_1x+b_1$ where $a,b,a_1,b_1\in\mathbb{R}$ and equation $P(x)\cdot Q(x)=0$ has at most one real root then prove that the claims given below are true.
... | When it says both have at most one real root, it means $a^2-4b\le0$ and $a_1^2-4b_1\le0$.
Also, both inequalities cannot be equality, because they will have two real roots, or if both inequalities are equality, then $a=a_1$ and $b=b_1$.
Then $a+b+1\ge a+a^2/4+1=(a/2+1)^2\ge 0$
BUT
Because it can be $a=a_1=-2$ and $b=b_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4532738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solve the polynomial $x^6+x^5+x^4-2x^3-x^2+1=0$ in exact form
Try to reduce the degree of the polynomial $$P(x)=x^6+x^5+x^4-2x^3-x^2+1$$ by algebraic ways and find the possible solution method to $P(x)=0$.
The source of the problem comes from a non-english algebra precalculus workbook. It is a workbook for those who... | The trick $x+\frac1x=t$ requires the polynomial to be a palindrome, so that is clearly not the case here. However that's not the only substitution possible to simply. Here, lets try something similar to what you have already done, persisting a bit more:
$
\begin{align}
\frac{P(x)}{x^4} &= x^2+x+1-\frac2x-\frac1{x^2}+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4534055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Compute the area of Quadrilateral $ABCD$ As title suggests, the question is to solve for the area of the given convex quadrilateral, with two equal sides, a side length of 2 units and some angles:
I have solved the problem with a synthetic geometric approach involving some angle chasing. However, I believe my solution... |
Let the intersection of the two diagonals be $O$.
And let $ AO = x , OC = y, AD = z $
Then it follows from the law of sines that
$ DO = a x \hspace{25pt}$ where $ a = \dfrac{\sin(105^\circ)}{\sin(30^\circ) } $
and
$ OB = b y \hspace{25pt}$ where $ b = \dfrac{\sin(75^\circ)}{\sin(60^\circ)} $
Apply the law of cosines t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4536997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
find two primes less than $500$ that divide $20^{22}+1$
Find two primes less than $500$ that divide $20^{22}+1$.
Note that $20^2+1=401$ divides the required number (since for any integer a, if k is odd, then $a+1$ divides $a^k+1$). Also, any prime dividing the given number must be congruent to $1$ modulo $4$ since $-... | Suppose a prime $p$ (necessarily odd) divides $20^{22} + 1$. Then $20^{22} \equiv -1 \bmod p$ so $20^{44} \equiv 1 \bmod p$ and it follows that the order of $20 \bmod p$ divides $44$ but does not divide $22$. This means it must be divisible by $4$, so must be either $4$ or $44$. If $20^4 \equiv 1 \bmod p$ then $20^{22}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4537818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
The limit $\lim_{n\to \infty}\frac{e+e^{\frac{1}{2}}+e^{\frac{1}{3}}+\ldots+e^{\frac{1}{n}}}{n}$ $$\lim_{n\to \infty}\frac{e+e^{\frac{1}{2}}+e^{\frac{1}{3}}+\ldots+e^{\frac{1}{n}}}{n}$$ is equal to
(a) $0$
(b) $1$
(c) $e$
(d) none of these
How should I deal with such limits? What I know $e >1$, so $$e+e^{\frac{1}{2}}+e... | Let's take @Gary:'s hint but in a pedestrian way:
First of all, we have Bernoulli's inequality
$$(1 + x)^n > 1 + n x$$
for $x > 0$, and $n > 1$. Therefore we conclude
$$1+ x > ( 1 + n x)^{\frac{1}{n}}$$
or
$$1 + \frac{y}{n} > (1+y)^{\frac{1}{n}}$$
for $y> 0$ and $n> 1$ natural. In particular
$$1 + \frac{e-1}{n} > e^{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4540703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Expectation of $E(XY^2)$ where $X$ and $Y$ are random events related to drawing balls from a vase. The Question:
a vase has 2 white and 2 black balls in it, we remove balls from the vase randomly one by one without returning them. $X=$ the number of balls that we removed until both white balls were removed (incl the l... | There are $6$ equiprobable permutations of the set $\{w,w,b,b\}$, and for each one, we compute the values of $X$ and $Y$ and $XY^2$ as follows:
$$\begin{array}{c|c|c|c}
\text{Outcome} & \color{red}{X} & \color{blue}{Y} & XY^2 \\
\hline
(w,\color{red}{w},\color{blue}{b},b) & 2 & 3 & 18 \\
(w,\color{blue}{b},\color{red}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4544541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is there a closed form expression for $\sum_{1\le i_1 \le i_2 \le \dots \le i_k \le n} i_1 i_2 \dots i_k$? These sums showed up in a probability problem I was working on. They're not quite the Stirling numbers of the first kind since it's possible to have e.g. $i_1 = i_2$. Denoting the sum by $(k\mid n)$ we have the re... | We get from first principles the generating function where we mark
multisets containing some number of instances of $q$ giving the factor
being contributed and the exponent of $z$ how often it occurs:
$$[z^k] \prod_{q=1}^n (1+qz+q^2z^2+q^3z^3+\cdots)
= [z^k] \prod_{q=1}^n \frac{1}{1-qz}
\\ = \frac{1}{n!} [z^k] \prod_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4544787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 2
} |
Angle Between Two Vectors Discrepancy Here is something that is bugging me: Consider the vectors
$$ \mathbf{v} = 2\mathbf{i} - 4\mathbf{j} + 4\mathbf{k} \qquad \text{ and } \qquad \mathbf{w} = 2\mathbf{j} - \mathbf{k}. $$
We have two different formulas for the angle $\theta$ between these vectors:
$$ \theta = \cos^{-1}... | The angle between two vectors should always be a positive number between $0$ and $\pi$ inclusive, so if the cosine of the angle is a negative number, the value should be between $\pi/2$ and $\pi$ inclusive.
Your numbers $-2/\sqrt5$ and $1/\sqrt5$ clearly obey $\cos^2\theta+\sin^2\theta=1$ and thus form a valid sine and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4546927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
easy way to calculate the limit $\lim_{x \to 0 } \frac{\cos{x}- (\cos{x})^{\cos{x}}}{1-\cos{x}+\log{\cos{x}}}$ I have been trying to use L'Hôpital over this, but its getting too long, is there a short and elegant solution for this?
The Limit approaches 2 according to wolfram.
| How to solve
$$
\begin{align*}
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{\cos(x) - \cos(x)^{\cos(x)}}{1 - \cos(x) + \log(\cos(x))}\\
\lim_{{x} \to {0}} f(x) &= \frac{\cos(0) - \cos(0)^{\cos(0)}}{1 - \cos(0) + \log(\cos(0))}\\
\lim_{{x} \to {0}} f(x) &= \frac{1 - 1^{1}}{1 - 1 + \log(1)}\\
\lim_{{x} \to {0}} f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4547927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Halmos Finite-Dimensional Vector Spaces Sec. 36 Ex. 5 on invertible transformations The exercise
If $A, B, C, D$ are linear transformations (all on the same vector space), and if both $A+B$ and $A-B$ are invertible, then there exist linear transformations $X$ and $Y$ such that
$$AX + BY = C$$
$$BX + AY = D$$
What I t... | We can apply Gaussian elimination reasonably well. The equations:
$$AX + BY = C \tag{1}$$
$$BX + AY = D \tag{2}$$
are equivalent to:
$$AX + BY = C \tag{1}$$
$$(A + B)X + (A + B)Y = C + D. \tag{3}$$
As you can see, $(3) = (1) + (2)$. We can recover $(2)$ by computing $(3) - (1)$. As $A + B$ is invertible, $(3)$ is equiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4549932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Laurent series expansion of $\frac{1}{z(z-2)^2}$ at $z=2$ Im struggling a bit with Laurent series expansions, and I am stuck on the following exercise:
Find a Laurent series expansion with center $z=2$ of $f(z)=\frac{1}{z(z-2)^2}$
My solution so far is not a lot:
I've found that there will be one Laurent series at the ... | Using partial fraction expansion, we can write
$$\begin{align}
\frac1{z(z-2)^2}=\frac1{4z}-\frac1{4(z-2)}+\frac1{2(z-2)^2}
\end{align}$$
To obtain the Laurent series for $|z-2|<2$, we write
$$\begin{align}
\frac1{z(z-2)^2}&=\frac1{4(z-2+2)}-\frac1{4(z-2)}+\frac1{2(z-2)^2}\\\\
&=\frac18 \frac1{1+\frac{z-2}2}-\frac1{4(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4550619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find the total number of roots of $(x^2+x+1)^2+2=(x^2+x+1)(x^2-2x-6)$, belonging to $(-2,4)$.
Find the total number of roots of $(x^2+x+1)^2+2=(x^2+x+1)(x^2-2x-6)$, belonging to $(-2,4)$.
My Attempt:
On rearranging, I get, $(x^2+x+1)(3x+7)+2=0$
Or, $3x^3+10x^2+10x+9=0$
Derivative of the cubic is $9x^2+20x+10$
It is z... | Since $ \ x^2 + x + 1 \ = \ \left(x + \frac12 \right)^2 + \frac34 \ > \ 0 \ $ for all real numbers, we might ask under what circumstances $ \ (x^2+x+1)^2+2 \ > \ (x^2+x+1)(x^2-2x-6) \ \ , \ $ since the product on the right side can plainly take on negative values. Dividing out a factor of $ \ (x^2+x+1) \ $ produces
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4553039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Solve the system $ \left\lbrace \begin{array}{ccc} \sqrt{1-x} + \sqrt{1-y} &=& \frac{y-x}{2} \\ x+y &=& 2xy \end{array}\right. $ I have to solve the following system:
$$
\left\lbrace \begin{array}{ccc}
\sqrt{1-x} + \sqrt{1-y} &=& \frac{y-x}{2} \\
x+y &=& 2xy
\end{array}\right.
$$
I've showed that it is equivalent to
$... | $$
\left\lbrace \begin{array}{ccc}
\sqrt{1-x} - \sqrt{1-y} &=& 2 &(1)\\
x+y &=& 2xy &(2)
\end{array}\right.
$$
From 1, get squared both sides two times, replace all (x + y) by 2xy, you will recieve a equation of (xy).
Find the appropriate of (xy) and then solve the (2) equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4556169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Combinatoric identity from a random walking problem I tried to prove the following equation and got stuck:
$$\sum_{k=1}^{n-r+1}\left[\binom{2k-1}{k}-\binom{2k-1}{k+1}\right]\binom{2n-2k+1}{n-k+r}=\binom{2n+1}{n-r}.$$
For whom are interested in its background:
Let $S_n$ be a symmetric random walking on the line. We call... | We seek to verify that
$$\sum_{k=1}^{n-r+1}
\left[{2k-1\choose k} - {2k-1\choose k+1}\right]
{2n-2k+1\choose n-k+r} = {2n+1\choose n-r}.$$
Note that for the square bracket term with $k\ge 1$ we get a Catalan
number, so the LHS becomes
$$\sum_{k=1}^{n-r+1} \frac{1}{k+1} {2k\choose k}
{2n-2k+1\choose n-r+1-k}.$$
This bec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4558007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How can we generalize factorisation of $(a+b)^n-(a^n+b^n)$
How can we generalize factorisation of
$$(a+b)^n-(a^n+b^n)\,?$$
where $n$ is an odd positive integer.
I found the following cases:
$$(a+b)^3-a^3-b^3=3ab(a+b)$$
$$(a+b)^5-a^5-b^5=5 a b (a + b) (a^2 + a b + b^2)$$
$$(a+b)^7-a^7-b^7=7ab(a+b)(a^2+ab+b^2)^2$$
Exap... | For any case obviously you have $a$ and $b$ in each term since you've removed both the $a$-only and the $b$-only terms from $(a+b)^n$. Also $a=-b$ is always a root, so you'll always get an $(a+b)$ term. So $ab(a+b)$ is a given for any odd $n$.
The roots of $a^2+ab+b^2$ are $a={1\over2}(-b\pm{\sqrt{b^2-4b^2}})=-b\omega,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4563475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Evaluating $\int{\sqrt{x^2+4x+13}\, dx}$ I was trying to calculate with $\sinh$ :
$$\begin{align}I&=\int{\sqrt{x^2+4x+13}\, dx}\\&=\int{\sqrt{(x+2)^2+9}\, dx}\end{align}$$
Now with $x+2=3\sinh(u)$, $dx=3\cosh(u)\,du$
$$\begin{align}I&=3\int{\left(\sqrt{9\sinh^2(u)+9}\, \right)\cosh(u)\, du}\\&=9\int{\cosh(u)\sqrt{\sinh... | Since $\frac12\sinh(2u)=\sinh(u)\cosh(u)=\sinh(u)\sqrt{1+\sinh^2(u)}$, you have
\begin{align}
\frac12\sinh(u)&=\frac{x+2}3\sqrt{1+\left(\frac{x+2}3\right)^2}\\
&=\frac19(x+2)\sqrt{x^2+4x+13}.
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4565199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Prove: $\int_{-1}^1 x^n P_n(x) dx = \frac{2^{n+1}n!^2}{(2n+1)!}$ I'm trying to prove:
$$\int_{-1}^1 x^n P_n(x) dx = \frac{2^{n+1}n!^2}{(2n+1)!}$$
My attempt consisted in applying the Rodrigues formula:
$$\int_{-1}^1 x^n P_n(x)dx = \dfrac{1}{2^n n!} \int_{-1}^1 x^n \dfrac{d^n}{dx^n}(x^2-1)^ndx$$
and integrating by parts... | A method based upon the series for $P_{n}(x)$ is seen by the following.
Using
$$ P_{n}(x) = \sum_{k=0}^{\lfloor{n/2}\rfloor} \frac{(-1)^k \, \left(\frac{1}{2}\right)_{n-k} \, (2 x)^{n-2 k}}{k! \, (n-2 k)!} $$
then
\begin{align}
I &= \int_{-1}^{1} x^n \, P_{n}(x) \, dx \\
&= \sum_{k} a_{k}^{n} \, 2^{n-2 k} \, \int_{-1}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4566824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find the volume of a bounded region rotated about the $x$-axis Question
A region is bounded by the $x$-axis, $y=x^2-9$ and $y=x^2-4$, where the region looks similar to a u-shape. The region is then rotated about the $x$-axis. Find the volume of the solid formed.
My Working
We can see that $y=x^2-9$ intersects the $x$-a... | After looking at CyclotomicField's comment, it seems that the teacher's answer is right and that I have just made a silly mistake, as shown below:
\begin{align}
V&=2\pi\left(49+\frac{211}{5}+\frac{64}{3}\right)\\
&=2\pi\cdot\frac{735+633+320}{15}\\
&=2\pi\cdot\frac{1688}{15}=\frac{3376}{15}\pi
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4568852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find minimum n :$ \prod_{k=1}^n \frac{\arcsin\left( \frac{9k+2}{\sqrt{27k^3+54k^2+36k+8}}\right)}{\arctan \left( \frac{1}{\sqrt{3k+1}} \right)}>2000$ We've got to find the minimum value of n for which
$$ \prod_{k=1}^n \frac{\arcsin\left( \frac{9k+2}{\sqrt{27k^3+54k^2+36k+8}}\right)}{\arctan \left( \frac{1}{\sqrt{3k+1}}... | It just clicked a few minutes after I posted it
We've got:
$$\prod_{k=1}^n \frac{\arcsin\left( \frac{9k+2}{(3k+2)^{\frac{3}{2}}}\right)}{\arcsin \left( \frac{1}{(3k+2)^{\frac{1}{2}}} \right)}>2000$$
let $\sin(\theta)=\frac{1}{(3k+2)^{\frac{1}{2}}}$
now consider
$$\sin(3\theta)=\frac{3}{(3k+2)^{\frac{1}{2}}}-\frac{4}{(3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4572253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Extracting sequence from generating function I was provided the following generating function, and was unsure how to use it. I have never seen an example where the function “involved” itself.
The generating function is
$F(z)^8$
Where
$$F(z)=z+z^6 F(z)^5+z^{11} F(z)^{10}+z^{16} F(z)^{15}+z^{21} F(z)^{20}$$
Any help appr... | Use Lagrange inversion. From the functional equation that $F(z)$ satisfies, we get, by multiplying through by $z$,
$$
zF(z)=z^2\left(1+(zF(z))^5+(zF(z))^{10}+(zF(z))^{15}+(zF(z))^{20}\right).
$$
Thus,
$$
\frac{z}{1+z^5+z^{10}+z^{15}+z^{20}}\circ(zF(z))=z^2,
$$
so
$$
zF(z)=\left(\frac{z}{1+z^5+z^{10}+z^{15}+z^{20}}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4576868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Isolating $θ$ in $a \sin^2 \theta + b \sin \theta + c + d \cos^2 \theta + e \cos \theta + f = 0$. I have an equation that follows this form:
$a \sin^2 \theta + b \sin \theta + c + d \cos^2 \theta + e \cos \theta + f = 0$ where $a,b,c,d,e,f \in \mathbb{R}$ and $a\neq 0, d\neq 0$. I am trying to isolate $θ.$
How can this... | You can convert this expression to a $4$-th order polynomial with the Expotential representations of the trigonometric functions and substitution
these with $t$. You can now solve this equation with the polynomial of degree $4$ using a solution formula or equivalent transformation after t and then resubstitute, so that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4577312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to solve the differential equation $y'=\frac{y-xy^2}{x+x^2y}$ $$y'=\frac{y-xy^2}{x+x^2y}$$
This is the equation I want to solve.
My idea is to substitute $xy$ with $u,$ $u=xy$ and $\frac{du}{dx}=xy'+y.$
So, the equation becomes
$y'=\frac{y(1-u)}{x(1+u)}.$
What I am struggling with is how to deal with the $x$ and $y... | The equation is $$y^{\prime}=\frac{y-x y^2}{x+x^2 y}$$
HINT:
We can re-write it as: $$-\frac{1}{y^2}\frac{dy}{dx}=\frac{\left(x-\frac{1}{y}\right)}{x+x^2 y}$$
Now letting $\frac{1}{y}=t \Rightarrow -\frac{1}{y^2}\frac{dy}{dx}=\frac{dt}{dx}$
$$\Rightarrow \frac{d t}{d x}=\frac{t(x-t)}{x(x+t)}=\frac{\frac{t}{x}\left(1-\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4577530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Prove that $(x^2−1) \bmod 8$ $\in \{ 0 , 3 , 7 \}, \forall x \in \mathbb{Z}$. It must be verified that for all $x \in \mathbb{Z}$ it holds that $x^2 - 1 \bmod{8} \in \{0, 3, 7\}$. First some definitions. Using the following theorem a definition for $\bmod$ is provided:
Theorem. For all $a \in \mathbb{Z}$ and $d \in \m... | *
*If $x = 4 k$ with $k \in \mathbb{Z}$ then :
$$x^2 - 1 = 16 k^2 - 1 = 16 k^2 - 8 + 7 \equiv 7 mod 8$$
*If $x = 4 k + 1$ with $k \in \mathbb{Z}$ then :
$$x^2 - 1 = 16 k^2 + 8 k + 1 - 1 = 16 k^2 + 8 k \equiv 0 mod 8$$
*If $x = 4 k + 2$ with $k \in \mathbb{Z}$ then :
$$x^2 - 1 = 16 k^2 + 16 k + 4 - 1 = 16 k^2 + 16 k ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4584496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Let $f(xy) = f(x)+ f(y) +\frac{x+y-1}{xy} \forall x,y>0$, $f'(1)=2$, $f$ is differentiable . Find $f(x)$. What I have gotten from $f(xy) = f(x)+ f(y) +\frac{x+y-1}{xy}$ is:
*
*$f(1)=-1$
2.$f(x)+f(\frac{1}{x}) =-(x+\frac{1}{x})$
The information I could get from $f'(1)=2$ is just:
$\lim_{h\to 0} { \frac{f(1+h)+1}{h}}=2... | Here is an answer along the lines requested by OP by taking a limit.
I assume the domain is $x > 0 $
I use an alternate, but equivalent, definition of the derivative involving a product to make use of the given property. It is valid since $xh \to x$ as $h \to 1$.
$$
\begin{align*}
f'(x) &= \lim_{h \to 1} \frac{f(xh) -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4591396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
What is the value of $f(1999)$? Let be a function $f \colon \Bbb R\to \Bbb R$ given from:
$$f(x)=x^3+\sqrt{x^6+1}+\frac{1}{x^3-\sqrt{x^6+1}}$$
What is the value of $f(1999)$?
To me, it immediately seemed strange that it was necessary to calculate $f(1999)$. With a calculator everything would come easy. Since this quest... | Your work seems correct to me. (this is stated in the comments that your work is correct)
The problem statement is essentially equivalent to:
Let $f:\mathbb R\rightarrow \mathbb R$,
$$f(x):=A(x)+B(x)+\frac {1}{A(x)-B(x)}$$
where, $A^2(x)-B^2(x)=-1$.
Find the value of $f(1999).$
$$
\begin{align}f(x):&=A(x)+B(x)+\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4592316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Calculation limit: $\lim\limits_{n\to\infty} \sum\limits_{k=0}^n \frac{1}{2^k-n}$
$$\lim_{n\to\infty} \sum_{k=0}^n \frac{1}{2^k-n}$$
My main issue with the sequence $a_k=\frac{1}{2^k-n}$ is that there will always be a value of $k$ for which the sequence will not be properly defined (at least if $n=2^k$ for some value... | Consider the subsequence of $n \not= 2^m$. Now it makes sense talking of the sequence and the limit (if it exists). Write $n=2^m + r$ with $r\in(0,2^m)$.
Then $\lfloor\log{n}\rfloor = m$ and $\lceil\log{n}\rceil = m+1$. Notice that $m\to \infty$ as $n \to \infty$, but the same is not true for $r$. So
$$
\sum_{k=0}^n \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4593766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.