Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
equation $\log_{2x}216=x$ Solve the equation $\log_{2x}216=x$ where $x$ is real.
My attempt,
By trial and error $x=3$ as $6^3=216$
Is there any systematic way to solve it?
| Noting that $$\log_{2x}(216) = \frac{\ln(216)}{\ln(2x)}$$ we can say \begin{align}\log_{2x}(216) &= x\\
\frac{\ln(216)}{\ln(2x)}&=x\\
\ln(216)&=x\ln(2x)\\
e^{\ln(216)}&=e^{x\ln(2x)}\\
e^{\ln(216)}&=e^{\ln((2x)^x)}\\
216&=(2x)^x
\end{align}
We can then note that $216=6^3$ and so \begin{align}6^3&=(2x)^x\\
(2\times 3)^3&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2274691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Distribute $a + b + c$ distinct balls into boxes $A, B, C$ such that $a$ balls, $b$ balls and $c$ balls go to boxes $A, B, C$
Distribute $a + b + c$ distinct balls into boxes $A, B, C$ such that $a$ balls, $b$ balls and $c$ balls go to boxes $A, B, C$, respectively, and show that the number of ways to do this is $\fra... | You say that "in all there are $10⋅9⋅8⋅7⋅6$ ways to put five balls into the box A". That is only true if the order in which we put the balls into the box matters. Clearly, it doesn't over here thus the correct computation would be $\displaystyle\binom{10}{5},$ that is $\dfrac{10⋅9⋅8⋅7⋅6}{5!}$ since we divide by the num... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2275298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Error in Polynomial Factoring
From Ramanujan's Notebooks IV:
Let $\alpha,\beta$ and $\gamma$ be the roots of$$x^3-ax^2+bx-1=0\tag1$$Now, choose cube roots such that $(\alpha\beta\gamma)^{1/3}=1$ and then let$$z^3-\theta z^2+\varphi z-1=0\tag2$$Denote the cubic polynomial with roots $\alpha^{1/3},\beta^{1/3},\gamma^{1/... | The last term on the left multiplies out to $3\theta\varphi(z^3-1)=3\theta\varphi z^3-3\theta\varphi$ which are intended to be the two middle terms from cubing the original right side. It should be $-3\theta\varphi z^3(\theta z^2-\varphi z)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2277000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Linear differential equation - inseparable? I need help with solving following linear differential equation:
$$y^\prime + \dfrac{3y}{x} = \dfrac{3}{x^4}$$
I have set the right side equal to $0$ and started solving:
$$y^\prime + \dfrac{3y}{x} = 0$$
$$y^\prime = - \dfrac{3y}{x}$$
$$\dfrac{dy}{dx}\times \dfrac{1}{y} = - ... | By multiplying both sides by the integrating factor $x^3$, we get
$$D(x^3y(x))=x^3y'(x) + 3 x^2 y(x) = \frac{3}{x}.$$
Hence
$$x^3y(x)=\int \frac{3}{x}\, dx=3\ln|x|+C,$$
that is
$$y(x)=\frac{3\ln|x|+C}{x^3}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2278198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Maximum value for $M^4$
$$M=|\sqrt{x^2+4x+5}-\sqrt{x^2+2x+4}|$$
Find the maximum value of $M^4$.
I think it could have a geometric solution. Because it looks like the difference between two points formula. Please help.
| $$\sqrt{x^2+4x+5}=\sqrt{(x+2)^2+1^2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)$$
$$\sqrt{x^2+2x+4}=\sqrt{(x+1)^2+(\sqrt{3})^2}~~~~~~~~~~~~~~~~~~~~~~~~(2)$$
$(1)$ is the distance of $(x,0)$ to $(-2,1)$ or $(-2,-1)$
$(2)$ is the distance of $(x,0)$ to $(-1,\sqrt{3})$ or $(-1,-\sqrt{3})$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2278884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Parallel Vectors and Coefficients of $\mu$ I have a question that states the following
Given that c = 3i + 4j and d = i - 2j, find μ if μc + d is parallel to i + 3j
So I wrote the following
$$ c = \begin{pmatrix} 3\\ 4\\ \end{pmatrix}$$
$$ d = \begin{pmatrix} 1\\ -2\\ \end{pmatrix}$$
I then wrote
$$ \mu \textbf{c} +... | Using your notation, we have a vector
$$\mathbf a= \begin {pmatrix}a_1\\a_2 \end{pmatrix}=\begin {pmatrix}1\\3 \end{pmatrix}$$ so a vector parallel to $\mathbf a$ is of the form
$$
\mathbf b =\begin {pmatrix}b_1\\b_2 \end{pmatrix}=\begin {pmatrix}b_1\\3b_1 \end{pmatrix}
$$
Apply this to
$$ \mathbf b=\begin {pmatrix}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Find integer part of the maximum value of $x^4y+x^3y+x^2y+xy+xy^2+xy^3+xy^4$ for $x+y=3$
If $M$ is the maximum value of $x^4y+x^3y+x^2y+xy+xy^2+xy^3+xy^4$. Subject to $x+y=3$. Find the value of $\lfloor M \rfloor$.
I think the maximum of $M$ occurred when $x=y=3/2$. And I just put the value of that in M and found 34.... | After substitution $y=3-x$ we get:
$$M=-11x^4+66x^3-139x^2+120x$$
$$M'(x)=2(3-2x)(11x^2-33x+20),$$
which gives a maximal value $\frac{400}{11}$ for $x=\frac{3+\sqrt{\frac{19}{11}}}{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2282911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
$\frac{(10^4+324)(22^4+324)\cdots(58^4+324)}{(4^4+324)(16^4+324) \cdots (52^4+324)}$
From AIME 1987, compute $$\frac{(10^4+324)(22^4+324)\cdots (58^4+324)}{(4^4+324)(16^4+324) \cdots (52^4+324)}$$
So basically the way used to solve this is by Sophie Germain's Identity which is $a^4+4b^4=(a^2+2b^2-2ab)(a^2+2b^2+2ab)$
... | (Without using Sophie Germain's identity, per OP's request.)
The given expression is $\,f(7, 5)\,$ where $\,\displaystyle
f(x, n) = \prod_{k=0}^{n-1} \frac{(x+12k+3)^4+324}{(x+12k-3)^4+324}\,$.
Taking each term in turn, let $\,y=x+12k\,$ then the fraction can be written as $\,\displaystyle\frac{(y+3)^4+324}{(y-3)^4+324... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2286733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Cubic equation with unknown coefficients given roots I was given this equation. $x^3 + 3px^2 + qx + r=0$. The roots are $1, -1$, and $3$.
Ive tried dividing the equation by $(x-1)$ to get a quadratic to make it easier for me. But that ended up really badly.
I also inputted the different roots into the equation to get ... | We can compare coefficients when we multiply all the roots in factor form:
$$\begin{align}(x-1)(x+1)(x-3) &= x^3-3x^2-x+3\\ &=x^3 + \underset{p}{3(-1)}x^2+\underset{q}{(-1)}x+\underset r{(3)}.\end{align}$$
We then check the roots:
1: $$(1)^3 - 3(1)^2 - 1 + 3 = 1-3-1+3 = 0$$
3: $$(3)^3 - 3(3)^2 - 3 + 3 = 3^3 - 3^3 + 0 =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2287301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Find the minimum value of $xy+yz+xz$ Question:
Find the minimum value of $xy+yz+xz$, given that $x,y,z$ are real and $x^2+y^2+z^2=1$
My attempt,
Since $(x+y+z)^2=x^2+y^2+z^2+2(xy+xz+yz)$
$=1+2(xy+xz+yz)$
$(x+y+z)^2\geq0$
So that $1+2(xy+xz+yz)\geq0$
$(xy+xz+yz)\geq -\frac{1}{2}$
So the minimum value is $-\frac{1}{2}... | You can also solve this problem using the method of Lagrange multipliers:
Note that $f(x,y,z) = xy + yz + xz$ is a $C^{\infty}$ function. We will use the method of Lagrange multipliers to determine the maximum/minimum of $f(x,y,z)$ given the restriction $g(x,y,z)= x^2 + y^2 + z^2 = 1$.
Therefore, because of Lagrange mu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2287522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving $\int_{0}^{\pi/2}\left[2\cos\left({x\over 2}\right)-x\sin\left({x\over 2}\right)\right]\ln\left[2\cos^2\left({x\over 2}\right)\right] dx...$ Proposed:
$$\int_{0}^{\pi/2}\left[2\cos\left({x\over 2}\right)-x\sin\left({x\over 2}\right)\right]\ln\left[2\cos^2\left({x\over 2}\right)\right]\mathrm dx=\color{green}{(... | Hint. One may write
$$
\begin{align}
&\int_{0}^{\pi/2}2\cos\left({x\over 2}\right)\ln\left[2\cos^2\left({x\over 2}\right)\right]\mathrm dx
\\\\&=\int_{0}^{\pi/2}2\cos\left({x\over 2}\right)\ln\left[2\left(1-\sin^2\left({x\over 2}\right)\right)\right]\mathrm dx
\\\\&=4\int_{0}^{\sqrt{2}/2}\ln\left[2\left(1-u^2\right)\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2287825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Generating functions ( recurrence relations ) Find $a_n$ using Generating Functions : $a_n = -a_{n-1} + 2a_{n−2}$, $n\ge2$ and $a_0 = 1$, $a_1 = 2$.
Approach : So I will form a characteristic equation $ r^2 + r - 2 = 0$ whose roots are $r_1 = -2$, $r_2 = 1$.
So my general solution is $a_n = α_1r_1^n + α_2r_2^n$.
$a_n =... | Define the generating function:
$\begin{equation*}
A(z)
= \sum_{k \ge 0} a_k z^k
\end{equation*}$
Write the recurrence with no subtraction in indices, multiply by $z^n$, sum over $n \ge 0$:
$\begin{equation*}
\sum_{n \ge 0} a_{n + 2} z^n
= - \sum_{n \ge 0} a_{n + 1} z^n
+ 2 \sum_{n \ge 0} a_n z^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2288360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Simplify algebraic expression with radicals: $\frac{1- ax}{1+ ax} \cdot \sqrt\frac{1+bx}{1-bx}$ I got stuck trying to simplify this roots forest:
$\frac{1- ax}{1+ ax}*\sqrt\frac{1+bx}{1-bx}$
where x= $\sqrt{\frac{2a}{b}-1}$
So it is:
$\frac{1- a\sqrt{\frac{2a}{b}-1}}{1+ a\sqrt{\frac{2a}{b}-1}}*\sqrt\frac{1+b\sqrt{... | HINT: we get
$$1-a\frac{\sqrt{2a-b}}{\sqrt{b}}=\frac{\sqrt{b}-a\sqrt{2a-b}}{\sqrt{b}}$$ for $b>0$
I have got the following result
$$\frac{\sqrt{b}-a\sqrt{2a-b}}{\sqrt{b}+a\sqrt{2a-b}}\cdot \frac{\sqrt{1+\sqrt{b}\sqrt{2a-b}}}{\sqrt{1-\sqrt{b}\sqrt{2a-b}}}$$
if $$2a-b\geq 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2288760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Can anyone help to proof convergence and find the sum of such series? May be correct my mistakes. $$1-\frac{1}{\sqrt{10}}-\frac{1}{10}+\frac{1}{10\sqrt{10}}-\frac{1}{10^{2}}-\frac{1}{10^{2}\sqrt{10}}+\cdot\cdot\cdot$$
I personally have such an idea: try to make geometric series like this
$$1-\frac{1}{\sqrt{10}}(1-\frac... | The identity of a series is often ambiguous when only a few example terms are given without a general rule for generating the rest.
I assume that the general rule in this case is that the magnitude of each term is $\newcommand{r}{{\sqrt{10}}} 1/\r$ times the magnitude of the previous term, while the signs repeat every ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2289659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
How to calculate this surface area? (portion of a cylinder inside a sphere ) The surface area of the portion of the cylinder $x^2+y^2=8y$ located inside of the sphere $x^2+y^2+z^2=64$
I'm stuck, so any tip will be helpful
Thanks in advance!
| $A = \iint dS$
$S: x^2 + y^2 = 8y$
Convert to cylindrical.
$x = r\cos\theta\\
y = r\sin\theta\\
z = z$
Plug these into the equation of the cylinder.
$r = 8\sin\theta$
And substitute back for parameterization of the surface
$x = 8\sin\theta\cos\theta = 4\sin 2\theta\\
y = 8\sin^2\theta = 4 - 4\cos 2\theta\\
z = z$
$dS =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2292416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Combinatorics: How many distinct binary strings of length 16 are possible if the zeroes must appear in groups of even number? I am trying to solve two combinatorics problems. May you help me?
First problem,
How many distinct binary strings of length 16 are possible if the zeroes must appear in groups of even number?
I ... | In order to count the number of strings of length $16$ having even runs of $0$s resp. pairs of $0$s we consider words
with no consecutive equal characters at all.
These words are called Smirnov words or Carlitz words. (See example III.24 Smirnov words from Analytic Combinatorics by Philippe Flajolet and Robert Sedgewi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2292605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Solving $|x^2-2x|+|x-4|>|x^2-3x+4|$
How do I solve $|x^2-2x|+|x-4|>|x^2-3x+4|?$
I can see the difference of the two terms on the left gives the term on the right. Now,what should I do? Is there any general method for solving $$|a|+|b|>|a-b|?$$
Thanks for any help!!
| Note that $|x^2-2x|+|x-4|=|x^2-2x|+|-x+4|\ge|x^2-3x+4|$ with the equality holds if and only if $x^2-2x$ and $-x+4$ are of the same sign.
Therefore, $|x^2-2x|+|x-4|>|x^2-3x+4|$ if and only if $x^2-2x$ and $x-4$ are of the same sign. So we have
\begin{align*}
(x^2-2x)(x-4)&>0\\
x(x-2)(x-4)&>0\\
0<x<2 \quad\textrm{or}\qu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2293534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Calculate $\lim_{x\to a}(2-\frac{x}{a} )^{\tan( \frac{\pi x}{2a})}$
I would like to calculate
$$\lim\limits_{x\to a}\left(2-\dfrac{x}{a} \right)^{\tan\left( \dfrac{\pi x}{2a}\right)},\quad a \in\mathbb{R}^* \,\,\text{fixed} $$
we've
$$\left(2-\dfrac{x}{a} \right)^{\tan\left( \dfrac{\pi x}{2a}\right)}=e^{\tan\left... | We can also apply L'Hopitals directly to $\log\left(2-\frac{x}{a}\right)\tan \frac{\pi x}{2a}$ and then appeal to continuity of $\exp$. In particular we have $$\lim_{x\to a} \frac{\sin \left(\frac{\pi x}{2a}\right) \log\left(2 - \frac{x}{a}\right) }{\cos \left(\frac{\pi x}{2a}\right)} =\lim_{x\rightarrow a} \frac{\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2294071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 1
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Show that $\int_0^\infty \frac{\ln x}{(x^2+1)(x^2-1)}dx=\frac{\pi^2}{8}$ How do I show that: $\int_0^\infty \frac{\ln x}{(x^2+1)(x^2-1)}dx=\frac{\pi^2}{8}$ using contours and residues
My attempt:
I know that the singular points are $i,-i,-1,1,0$
consider $f(z)= \frac{\ln z}{(z^2+1)(z^2-1)}$
and the branch $|z|>0$, $0<\... | Observe that
\begin{eqnarray*}
\frac{1}{(x^2+1)(x^2-1)}=\frac{1}{2(x^2-1)} + \frac{-1}{2(x^2+1)}
\end{eqnarray*}
So the original integral splits into the following two integrals
\begin{eqnarray*}
I_1=\int_{0}^{\infty} \frac{ln(x)}{2(x^2+1)} \\
I_2=\int_{0}^{\infty} \frac{ln(x)}{2(x^2-1)} .\\
\end{eqnarray*}
Both of th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2295370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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$\alpha + \beta + \gamma = \pi$ , show that $\cos 2\alpha + \cos 2\beta + \cos 2\gamma + 2\cos\alpha \cos\beta \cos\gamma = 1$
$\cos 2\alpha + \cos 2\beta + \cos 2\gamma + 2\cos\alpha \cos\beta \cos\gamma = 1$
I really didn't know how to solve this problem and I am very unused to the utilization of trigonometric iden... | I think you mean the following problem.
Let $\alpha+\beta+\gamma=\pi$. Prove that:
$$\cos^2\alpha+\cos^2\beta+\cos^2\gamma+2\cos\alpha\cos\beta\cos\gamma=1$$
We need to prove that
$$\cos^2\alpha+\cos^2\beta-2\cos\alpha\cos\beta\cos(\alpha+\beta)=\sin^2(\alpha+\beta)$$ or
$$\cos^2\alpha+\cos^2\beta-2\cos\alpha\cos\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2295677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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From $a_{n+1}=(1+ \frac{1}{n})^n \cdot a_n$ to $a_n$ to be proven by induction
Find and Prove by induction an explicit formula for $a_n$ if $a_1=1$ and for $n \geq 1$
$$a_{n+1}=(1+ \frac{1}{n})^n \cdot a_n$$
Checking the pattern:
$$a_1=1$$
$$a_2= 2 \cdot 1$$
$$a_3= (\frac{3}{2})^2 \cdot 2 \cdot 1$$
$$a_4= (\frac{4}... | $a_5= (\frac{5}{4})^4 \cdot (\frac{4}{3})^3 \cdot (\frac{3}{2})^2 \cdot 2 \cdot 1
$
$\begin{array}\\
a_n
&=\prod_{k=2}^n (\frac{k}{k-1})^{k-1}\\
&=\dfrac{\prod_{k=2}^n k^{k-1}}{\prod_{k=2}^n (k-1)^{k-1}}\\
&=\dfrac{\prod_{k=2}^n k^{k-1}}{\prod_{k=1}^{n-1} k^k}\\
&=\dfrac{\prod_{k=2}^n k^{k-1}}{\prod_{k=1}^{n-1} kk^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2295749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Prove that the following are real numbers
*
*$$\frac{1}{z}+\frac{1}{\overline{z}}$$
*$$z^3\cdot\overline{z}+z\cdot\overline{z}^3$$
1.$$\frac{1}{z}+\frac{1}{\overline{z}}$$
$$\frac{\overline{z}}{z\cdot \overline{z}}+\frac{z}{z\cdot\overline{z}}$$
$$\frac{\overline{z}+z}{z\cdot \overline{z}}$$
$$\frac{2Re(z... | If we know $z + \overline z \in \mathbb R$ and $z*\overline z = |z|^2\in \mathbb R$ and $\overline{\overline z} = z$ and $\overline {z*w} = \overline z*\overline w$ we are pretty much done.
1) $\frac 1{\overline z} *\overline z = 1$
$\overline{\frac 1{\overline z}*\overline z} = \overline 1 = 1$
$\overline{\frac 1{\o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2296860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
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Proof for sum of product of four consecutive integers I had to prove that
$(1)(2)(3)(4)+\cdots(n)(n+1)(n+2)(n+3)=\frac{n(n+1)(n+2)(n+3)(n+4)}{5}$
This is how I attempted to do the problem:
First I expanded the $n^{th}$ term:$n(n+1)(n+2)(n+3)=n^4+6n^3+11n^2+6n$.
So the series $(1)(2)(3)(4)+\cdots+(n)(n+1)(n+2)(n+3)$ wil... | $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2297845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 1
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Proving Trigonometric Equality I have this trigonometric equality to prove:
$$\frac{\cos x}{1-\tan x}-\frac{\sin x}{1+\tan x}=\frac{\cos x}{2\cos^2x-1}$$
I started with the left hand side, reducing the fractions to common denominator and got this:
$$\frac{\cos x+\cos x\tan x-\sin x+\sin x\tan x}{1-\tan^2x}\\=\frac{\cos... | From the last line of your working, and using $\sin^2x+\cos^2x=1,$
$$\frac{\cos^2x}{\cos^3x-\cos x\sin^2x}=\frac{\cos x}{\cos^2x-\sin^2x}=\frac{\cos x}{\cos^2x-(1-\cos^2x)}=\frac{\cos x}{2\cos^2x-1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2299015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 7,
"answer_id": 6
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Characteristic curves : Why do I get different results? With a given $h(x)$ we want to solve $$xu_y-yu_x=u \\ u(x,0)=h(x)$$
I have solved it using two ways.
$$$$
First way:
For $x\neq 0$ we get $u_y-\frac{y}{x}u_x=\frac{u}{x}$.
We have that $$\frac{du}{ds}=\frac{du}{dx}\cdot \frac{dx}{ds}+\frac{du}{dy}\cdot \frac{... | (The two solutions are the same. The notation tends to obscure the identity of the solutions. You have $h(\pm \sqrt{x^2+y^2})$ and $\dfrac{2h(\pm \sqrt{x^2+y^2})}{\pi}$. Nevertheless, they are not the same function. We need to make the following stipulation: $\dfrac{2h(\pm \sqrt{x^2+y^2})}{\pi}=g(\pm \sqrt{x^2+y^2})$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2299148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Square root of Matrix $A=\begin{bmatrix} 1 &2 \\ 3&4 \end{bmatrix}$ Find Square root of Matrix $A=\begin{bmatrix} 1 &2 \\ 3&4 \end{bmatrix}$ without using concept of Eigen Values and Eigen Vectors
I assumed its square root as $$B=\begin{bmatrix} a &b \\ c&d \end{bmatrix}$$
hence
$$B^2=A$$ $\implies$
$$\begin{bmat... | @ Umesh shankar, this is exactly what should not be done (if you want to make progress)!!
Note that $A^2=trace(A)A-\det(A)I=5A+2I$. Since $B^2=A$ we deduce that $A,B$ commute and since $A$ is not a scalar matrix, $B$ is in the form $B=aI+bA$. Then $(aI+BA)^2=A$, that is $a^2I+b^2(5A+2I)+2abA=A$, that implies
$$a^2+2b^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2303997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Express the equation as a single fraction in its simplest form Express the equation
$$ 4 - \left( (x + 3) ÷ {\frac {x² + 5x + 6}{x - 2}} \right) $$
as a single fraction in its simplest form.
So far all I managed to do was factorise the numerator to get this $$(x + 3)(x + 2)$$
Where do I go from here?
| Well here is an answer for if the expression is as I have written it in my comment above. If my comment indeed has the wrong expression, then the steps are similar, but involves a closer look using the proper order of operations.
We first use the fact that $\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}$ where... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2307471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all positive integers $n \le 6$, such that the equation $a^n+b^n=c^n+n$ has solutions over the integers. Find all positive integers $n \le 6$, such that the equation $a^n+b^n=c^n+n$ has solutions over the integers.
My attempt:
For $n=1$, trivially there are integer solutions.
For $n=2$, we have the solutions
$$a=b... | $n=4$:
fourth powers are $\equiv 0$ or $1\pmod{8}$. There is no way to have $(0\text{ or }1)+(0\text{ or }1)\equiv (0\text{ or }1)+4\pmod 8$.
$n=5$:
Fifth powers are $\equiv 0$ or $1$ or $-1\pmod{11}$. Three such numbers combined cannot produce $5$, i.e., $a^5+b^5-c^5\equiv -3,-2,-1,0,1,2,3\not\equiv 5\pmod {11}$.
$n=6... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2309808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $\sin(5\theta)=1$, $0<\theta<2\pi$. Show that the roots of $16x^4+16x^3-4x^2-4x+1=0$ are $x=\sin{\frac{(4r+1)\pi}{10}}$, $r=0,2,3,4$. This is a very interesting problem that I came across. I know it's got something to do with trigonometry identities, polynomials and complex numbers, but other than that, I'm not t... | Think of the equation
$1=\sin 5\theta$
$=\sin3\theta cos2\theta+\cos3\theta sin2\theta$
$=(3\sin\theta-4\sin^3\theta)(1-2\sin^2\theta)+(4\cos^3\theta-3\cos\theta) 2\sin\theta \cos\theta$
$=(3\sin\theta-4\sin^3\theta)(1-2\sin^2\theta)+(4\cos^2\theta-3) 2\sin\theta (1-\sin^2\theta)$
$=(3\sin\theta-4\sin^3\theta)(1-2\sin^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2309998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to find the sum of this series: $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^2}{n^3+1}$? $$\frac{1^2}{1^3+1}-\frac{2^2}{2^3+1}+\frac{3^2}{3^3+1}-\frac{4^2}{4^3+1}+\cdots$$
in terms of summation i can write it as
$$S_{n}=\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^2}{n^3+1}$$
How to continue from this point?
used partial frac... | From partial fractions,
$$ \frac{n^2}{n^3 + 1} = \frac{1}{3(n+1)} + \frac{1}{3(n-r)} + \frac{1}{3(n-\overline{r})} $$
where $r$ is a root of $z^2 - z + 1$.
Now $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n +1} = 1 - \log(2) $$
while
$$ \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n - r} = \frac{1}{2} \Psi\left(1 - \frac{r}{2}\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2311081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Number Theory - Modular arithmetic with complex numbers So I was looking at a problem that asked me to find an $x$ s.t $x^3 = -1 \pmod{199}$ given that $14^2 = -3 \pmod{199}$.
To do this I was given the hint to look at complex numbers so I did the following
$$2(-1)^{\frac{1}{3}} - 1 = \sqrt{-3} $$
so then I can say tha... | $x^3 + 1 = (x + 1)(x^2 - x + 1)$ and note that $199$ is a Prime, so
$x + 1 = 0 \ \pmod{199}$ or $x^2 - x + 1 = 0 \ \pmod{199}$.
The hint $14^2 = -3 \ \pmod{199}$ to find the root of $x^2 - x + 1$ since:
$x^2 - x + 1 = (x - \frac{1}{2})^2 + \frac{3}{4} = 0 \ \pmod{199}$
then
$(2x - 1)^2 = -3 \ \pmod{199}$
p.s.
there ... | {
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"url": "https://math.stackexchange.com/questions/2311399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Solving recurrence relations of type $S_n-7S_{n-1}+10S_{n-2}=5\cdot 3^n$ I know how to solve this kind of equasions
$$S_n-7S_{n-1}+10S_{n-2}=5\cdot 3^n$$ $$S_0=0, S_1=1$$ for example...but when there is a constant (example:$(3^n+5))$ or $(5\cdot 3^n)$ i don't know how to solve it. Any tips.
| Since the characteristic polynomial is $z^2-7z+10 = (z-2)(z-5)$, the solutions of
$$ S_{n}-7 S_{n-1} + 10 S_{n-2} = 0 $$
have the form $S_n = \alpha 2^n+\beta 5^n$. By direct inspection a solution of
$$ S_{n}-7 S_{n-1} + 10 S_{n-2} = 5\cdot 3^n $$
is given by $S_n=-\frac{45}{2}\cdot3^n$, hence the set of solutions of t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $\int_{0}^{\infty}{x\over (1+x^2)^2}\cdot{\mathrm dx\over \tanh\left({\pi x\over 2}\right)}={\pi^2\over 8}-{1\over 2}?$ How may we show that
$$\int_{0}^{\infty}{x\over (1+x^2)^2}\cdot{\mathrm dx\over \tanh\left({\pi x\over 2}\right)}={\pi^2\over 8}-{1\over 2}\color{red}?\tag1$$
$u={x\over 2}\implies 2du=dx$... | Another approach is to use the partial fraction expansion of $\coth(z)$.
(See THIS QUESTION for derivations.)
$$\begin{align} & \int_{0}^{\infty} \frac{x}{(1+x^{2})^{2}} \coth \left(\frac{\pi x}{2} \right) \, dx \\ &= \int_{0}^{\infty} \frac{x}{(1+x^{2})^{2}} \left(\frac{2}{\pi x} + \pi x \sum_{n=1}^{\infty} \frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2315485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
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If corr(A,B) = x and corr(B,C) = y, what is corr(A,C)? If I know the correlation between two pairs of random variables (A,B) and (B,C), can I determine the correlation of the pair (A,C)? If not, can I at least constrain it to some range?
I'm interesting in generating a covariance matrix where certain pairwise correlati... | Unfortunately no, we cannot determine $\rho_{AC}$ given just $\rho_{AB}$ and $\rho_{BC}$. You can derive the theoretical bounds
\begin{align*}
\rho_{AC} \ge \max\{2(\rho_{AB} + \rho_{BC}) - 3, 2\rho_{AB}\rho_{BC} - 1\}
\end{align*}
Proof. Some notation. I let $\sigma_{AB} = \text{Cov}(A,B)$ and $\sigma_A^2 = \text{Var... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2316314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why does my parametric trigonometric function appears to be a polynomial? I was fooling around with $(\cos^2(t),\sin^2(at))$ with varying values of $a$, and found that if $a=3$ then $(\cos^2(t),\sin^2(3t))$ gives the graph of $y=-16x^3+24x^2-9x+1$ on the domain $[0,1]$
The calculator won't do parametrics but it just l... | We have $$\begin{align*}y=\sin^2 3t &= (3\sin t - 4\sin^3 t)^2 \\ &= \sin^2 t(3-4\sin^2 t)^2 \\& = (1-\cos^2 t)(3-4(1-\cos^2 t))^2 \\ & = (1-x)(4x-1)^2 \\ & =-16x^3+24x^2-9x+1\end{align*}$$
where we derived $\sin 3t = 3\sin t - 4\sin^3 t$ either by expanding $e^{3it}$ and taking imaginary parts or by expanding $\sin 3t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2316832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solve $\sqrt3 \cot^2 x - 2 \cot x - \sqrt3 = 0$ I thought it'd be nice to factor this as follows: $\sqrt3 \cot^2 x - 2 \cot x - \sqrt3 = 0 $, so: $(\sqrt3 \cot x + 1)(\cot x - \sqrt3) = 0$.
So there are 2 equations to solve:
*
*$\sqrt3 \cot x + 1 = 0$. I thought to multiply both sides by $\tan
x$, yielding: $\sq... | Assuming $\tan x\neq 0$, multiplying throughout by $\tan^2x$, moving terms to RHS, and putting $t=\tan x$ gives
$$\sqrt3\ t^2+2t-\sqrt3=0\\
(\sqrt3\ t-1)(t+\sqrt3)=0\\
t=\tan x=\frac 1{\sqrt3}, -\sqrt3\\
x=n\pi+\frac {\pi}6, n\pi-\frac {\pi}3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2318653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evalaute $\int_{-\alpha}^{\alpha}{1\over x}\sqrt{\alpha+x\over \alpha-x}\ln^n\left({\alpha+x\over \alpha-x}\right)\mathrm dx=(-\pi)^{n+1}F(n)$ Proposed:
$$\int_{-\alpha}^{\alpha}{x^k}\sqrt{\alpha+x\over \alpha-x}\ln^n\left({\alpha+x\over \alpha-x}\right)\mathrm dx=F(k,n)\tag1$$
Let $k=-1$ and $F(-1,n)=F(n)$
$$\int_{-\... | The comment of Simply Beautiful Art gives a big hint on how to proceed. I cannot give a general formula to (1), although it is possible albeit complicated.
Let $$I(k,t)=\int_{-1}^{+1}x^k\left(\frac{1+x}{1-x}\right)^t dx \quad\quad J(k,n)=\int_{-1}^{+1}x^k\sqrt{\frac{1+x}{1-x}}\ln^n\left(\frac{1+x}{1-x}\right)dx$$
makin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2319587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Lack of understanding of steps for second principle of finite induction for: $a^n -1 = (a-1)(a^{n-1}+a^{n-2}+a^{n-3}+\cdots+a+1)$ The problem i'm trying to solve is:
Use the second principle of finite induction to establish that:
$$a^n -1 = (a-1)(a^{n-1}+a^{n-2}+a^{n-3}+\cdots+a+1)$$
for all $n \ge 1$
I have foun... | I'm just going to do the induction step here, for $k \to k + 1$, to show that it can be done in a simpler and more transparent way.
So assume you already know that
$$a^k - 1 = a^{k-1} + a^{k-2} + \dots + a + 1.$$
Then, using the induction hypothesis on the second line, we have
\begin{align}
(a-1)(a^k + a^{k-1} + \dots... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2322833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to solve $\log_{4}(\sqrt{x^{4/3}})+3\log_{x}(16x)=7?$
How to solve $\log_{4}(\sqrt{x^{4/3}})+3\log_{x}(16x)=7?$
I've tried everything from brute force to doing base change but nothing works. I was wondering what was the best way in solving this?
| \begin{align}
\log_{4}(\sqrt{x^{4/3}})+3\log_{x}(16x)&=7\\
\left(\frac{1}{2}\right)\left(\frac{4}{3}\right)\log_{4}(x)+3\log_{x}(4^2)+3\log_{x}(x)&=7\\
\frac{2}{3}\log_{4}(x)+\frac{6}{\log_4x}+3&=7\\
(\log_4x)^2-6\log_4x+9&=0\\
\log_4x&=3\\
x&=64
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2323086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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If $f (x) +f'(x) = x^3+5x^2+x+2$ then find $f (x)$
If $f (x) +f'(x) = x^3+5x^2+x+2$ then find $f(x)$.
$f'(x)$ is the first derivative of $f (x)$.
I have no idea about this question, please help me.
| The hard way:
You can use the fact that
$$(e^xf(x))'=e^x(f(x)+f'(x)).$$
Then
$$(e^xf(x))'=e^x(x^3+5x^2+x+2).$$
Now, by integration (which can be performed by parts, integrating $e^x$),
$$e^xf(x)=\int e^x(x^3+5x^2+x+2)dx=e^x(x^3+2x^2-3x+5)+C$$ and
$$f(x)=x^3+2x^2-3x+5+Ce^{-x}.$$
[I doubt this is the method you are expec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2323688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 2
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Find $P$ such that $P^TAP = D$ where $D$ is a diagonal matrix.
Let $$A = \left(\begin{array}{cc} 2&3 \\ 3&4 \end{array}\right) \in
M_n(\mathbb{C})$$
Find $P$ such that $P^TAP = D$ where $D$ is a diagonal matrix.
How can I find $P$? I am doing Gauss but it does not work?$$A = \left(\begin{array}{cc|cc} 2&3&1&0\\ 3&4&... | Problem
Diagonalize the matrix
$$
\mathbf{A} =
\left[
\begin{array}{cc}
2 & 3 \\
3 & 4 \\
\end{array}
\right]
$$
Solution
Compute eigenvalues
The eigenvalues are the roots of the characteristic polynomial
$$
p(\lambda) = \lambda^{2} - \lambda \text{ trace }\mathbf{A} + \det \mathbf{A}
$$
The trace and determinant a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2324359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Trigonometric Equation : $\sin 96^\circ \sin 12^\circ \sin x = \sin 18^\circ \sin 42^\circ \sin (12^\circ -x)$ Please help solving this equation:
$\sin 96^\circ \sin 12^\circ \sin x = \sin 18^\circ \sin 42^\circ \sin (12^\circ -x)$
I used numerical method to solve it and got $x=6^\circ$ but I am not able to solve it ... | Using the same observation as @dantopa we get
\begin{eqnarray}
\sin(x) &=& \sin(12^\circ - x )\\
\sin(x) &=& \sin(12^\circ)\cos(x)-\cos(12^\circ)\sin(x)\\
\tan(x) &=& \sin(12^\circ)-\cos(12^\circ)\tan(x)\\
\tan(x)(1+\cos(12^\circ))&=&\sin(12^\circ)\\
\tan(x)&=&\frac{\sin(12^\circ)}{1+\cos(12^\circ)}\\
\tan(x)&=&\tan(6^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2324425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Laurent Expansion - Complex Analysis question Exercise :
Find the Laurent Expansion of the function :
$$f(z) = \frac{1}{(z-2i)(z^2 +4)}$$
around $z_0 = -2i$, in the biggest possible ring that includes the point $z=-2 + 2i$.
Attempt :
We have :
$$f(z) = \frac{1}{(z-2i)(z^2 +4)} = \frac{1}{(z-2i)(z + 2i)(z-2i)} = \frac{1... | You are on the right track. Just the last step should be revised a little.
The function $$f(z)=\frac{1}{(z-2i)^2}\cdot\frac{1}{(z + 2i)}$$ is to expand around the center $z_0=-2i$. Since there are poles at $z=2i$ and $z=-2i$ we have to distinguish two regions
\begin{align*}
D_1:&\quad 0<|z+2i|<4\\
D_2:&\quad |z+2i|>4... | {
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"url": "https://math.stackexchange.com/questions/2325055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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How to prove by induction that for $n$ $ \in \mathbb N $ , $ 2 n - 18 < n^2-8n +8 $? Question:
Prove for $n$ $ \in \mathbb N $ , $ 2 n\ -\ 18\ <\ n^2-8n\ +8 $
My attempt:
$ Base\ Case:\ n\ =\ 1,\ it\ holds. $
$I.H:\ Suppose\ 2k-18\ <\ k^2-8k+8,\ where\ k\ is\ a\ natural\ number.$
$ Then,\ \left(k+1\right)^2-8\left(k+... | Notice, $2n-18<n^2-8n+8 \implies n^2-10n+26=(n-5)^2+1>0$.
Let $(n-5)=u$, so that we have $u^2+1>0$.
Moving over the $1$ gives us $u^2>-1$. We know this is true because all squares of real (and natural) numbers are greater than $-1$.
Therefore, we have $(n-5)^2>-1$, and $(n-5)^2+1>0$, and $n^2-10n+26>0$.
Adding $2n-18$... | {
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"url": "https://math.stackexchange.com/questions/2325674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Series expansion of Elliptic integral $F(\sin^{-1}(1-x)|-1)$ where $0I understand the leading term $F(\pi/2\mid-1)$ would be $K(-1)$. I also want the next term contributing from $x$. Actually I could not expand $\sin^{-1}(1-x)$ (when $0<x\ll1$) around 1. Any help would be highly appreciated.
Thanks a lot.
| For more than the first term.
We can start using $$\frac d {dx}F\left(\left.\sin ^{-1}(1-x)\right|-1\right)=-\frac{1}{\sqrt{1-(1-x)^2} \sqrt{1+(1-x)^2}}\tag 1$$ and use Taylor expansions around $x=0$ $$\sqrt{1-(1-x)^2}=\sqrt{2} \sqrt{x}-\frac{x^{3/2}}{2 \sqrt{2}}-\frac{x^{5/2}}{16
\sqrt{2}}-\frac{x^{7/2}}{64 \sqrt{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2326743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Determine if $\lim_{(x,y)\to(0,0)} \frac{x^3y^4}{(x^4 + y^2)^2}$ exist What I tried:
Let $$\ f(x,y) = \frac{x^3y^4}{(x^4 + y^2)^2}$$
For points of the form$\ (x,0)$ then $\ f(x,0)=0$, similarly, for$\ (0,y)$ then $\ f(0,y)=0$, so lets suppose that:
$$\lim_{(x,y)\to(0,0)} \frac{x^3y^4}{(x^4 + y^2)^2} =0$$
So, for$\ ε>0$... | Using that for $y \neq 0$:
$$|\frac{x^3y^4}{(x^4+y^2)^2}| \leq |\frac{x^3y^4}{(0+y^2)^2}|=|x^3| \to 0$$
As a smaller denominator means larger number in magnitude. We easily conclude by squeeze theorem that the limit is $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2326934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
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Trigonometric Integral - Complex Integration Exercise :
If $a \in \mathbb R-\{0\}, |a|<1$, calculate the integral and show that :
$$\int_0^\pi \frac{\cos n\theta}{1 + a\cos \theta}d\theta = \frac{\pi}{\sqrt{1-a^2}}\Bigg(\frac{\sqrt{1-a^2} -1}{a} \Bigg)^n$$
Attempt :
Since $\cos \theta$ is even, we have :
$$\int_0^\pi \... | Let $z=e^{i\theta}$ and then
\begin{eqnarray}
&&\int_0^\pi \frac{\cos n\theta}{1 + a\cos \theta}d\theta\\
& =& \frac{1}{2}\int_{-\pi}^\pi \frac{\cos n\theta}{1 + a\cos \theta}d\theta\\
&=&\frac12\int_{|z|=1}\frac{\frac{z^n+\frac1{z^n}}{2}}{1+a\frac{z+\frac1z}{2}}\frac1{iz}dz\\
&=&\frac1{2i}\int_{|z|=1}\frac{z^{2n}+1}{z... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find high common factor of all possible $n$ If $n$ is an integer such that $2n + 1$ and $3n + 1$ are square numbers, find the highest common factor of all possible $n$.
Do we need to know exactly what all possible $n$ are?
| As $n=40$ gives us the squares $81=9^2$ and $121=11^2$, the desired number $d$ is a divisor of $40$, which limit sour search from above.
As $2n+1$ is an odd square, it os $\equiv 1\pmod 8$, hence $n\equiv 0\pmod 4$. But then $3n+1$ is also odd, hence $\equiv 1\pmod 8$, hence $8\mid n$.
Thus the only remaining possibili... | {
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"source": "stackexchange",
"question_score": "2",
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Triple integral using cylindrical coordinates with constraints The prompt is to find an iterated triple integral in cylindrical coordinate system where the surface S is defined as the solid cut out from the ball $x^2+y^2+z^2\leq 1$ by the half cone $z = \sqrt{(x^2 + y^2)}$ which represents the volume of the solid S.
So... | Visualizing the regions in question:
We do not want to integrate by $z$ last, as this would require two separate integrals. Noting that the volume we want has rotational symmetry (since both surfaces do) about the $z$-axis, the region on the $x$-$y$ or $r$-$\theta$ plane we want is a circle. So:
$$ \theta: 0 \text{ to... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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prove that : if $a, b \in \mathbb{R}^+$ : then : $a^2b^2(a^2+b^2-2)\geq (a^2+b^2)(ab-1)$ prove that : if $a, b \in \mathbb{R}^+$ : then :
$$a^2b^2(a^2+b^2-2)\geq (a^2+b^2)(ab-1)$$
$$a^4b^2+b^4a^2-2a^2b^2 \geq a^3b-a^2b+b^2a-a^2$$
$$a^4b^2+b^4a^2-2a^2b^2 \geq a^3b-a^2+b^3a-b^2$$
now what ?
| Let $a+b=2u$ and $ab=v^2$.
Thus, we need to prove that
$$v^4(4u^2-2v^2-2)\geq(4u^2-2v^2)(v^2-1)$$ or
$$v^4(2u^2-v^2-1)\geq(2u^2-v^2)(v^2-1)$$ or
$$2u^2v^4-v^6-v^4\geq2u^2(v^2-1)-v^4+v^2$$ or
$$2u^2(v^4-v^2+1)\geq v^6+v^2.$$
But $v^4-v^2+1>0$ and $u^2\geq v^2$ because it's just $(a-b)^2\geq0$.
Thus, it's enough to prove... | {
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"source": "stackexchange",
"question_score": "1",
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Solve $x^2 + (y-1)^2 + (x-y)^2 - \frac{1}{3} = 0.$ I came across an interesting equation with two variables $x,y\in\mathbb{R}$, $x^2 + (y-1)^2 + (x-y)^2 - \frac{1}{3} = 0.$
This, once expanded, can be simplified to $3x^2 - 3xy + 3y^2 - 3y + 1=0.$
How can one proceed to solve it algebraically? The solution according to ... | Consider the curve
$3x^2-3xy+3y^2-3y+k=0$, which can be written as
$$x^2-xy+y^2-y+\frac k3=0\tag{1}$$
which is an ellipse according to the discriminant criterion.
As the coefficients of $x^2, y^2$ are equal, this implies a rotation of $\frac \pi 4$.
After some manipulation we find that equation $(1)$ can be written... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Range in an Interval I want to find the range of the function, $$f(x) = {{x + 1} \over {{x^2} + 1}}\,\,\;{\rm{when }}\;x \in \left[ { - 1,1} \right].\;\;\;$$
I have isolated $x$ as - $$x = {{1 \pm \sqrt {1 - 4y(y - 1)} } \over {2y}}$$
But I am unable to solve this inequality, $$ - 1 \le {{1 \pm \sqrt {1 - 4y(y - 1)} } ... | Here is a method that avoids calculus completely.
Write $y = x+1$, then $\frac{x+1}{x^2+1} = \frac{y}{y^2-2y+2}$. Then note that $$\frac{y^2-2y+2}{y} = y-2+\frac{2}{y}.$$ Keeping in mind that the range of $y$ is $[0, 2]$, $y+\frac{2}{y}$ approaches $\infty$ as $y \to 0$, and the minimum value over positive $y$ is $2\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2329879",
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"source": "stackexchange",
"question_score": "2",
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$\sin(x) = \sin(x − \frac\pi3)$, solve for $x$ on interval $[-2\pi, 2\pi]$ According to the answer sheet:
$\sin(x) = \sin(x -\frac\pi3)$ gives:
$x = x-\frac\pi3 + k \cdot 2\pi$ or $x = \pi-(x-\frac\pi3) + k \cdot 2\pi$
^ How did they go from $\sin(x) = \sin(x-\frac\pi3)$ to the equations above?
Thanks in advance!
| There also exists a (seemingly) different approach. Recall that $$\sin x-\sin y =2\sin\frac{x-y}{2}\cos\frac{x+y}{2},$$
hence reforming your equation we obtain \begin{align}
0&=\sin x-\sin\left(x-\frac{\pi}{3}\right)\\&=2\sin\frac{x-\left(x-\frac{\pi}{3}\right)}{2}\cos\frac{x+\left(x-\frac{\pi}{3}\right)}{2}\\
&=2\sin\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2332378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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LU decomposition- Suppose we have to find LU decomposition of
$A = \begin{bmatrix}
3 & 1 & -1 \\
1 & 0 & 1 \\
4 & 2 & 2
\end{bmatrix}$
I correctly found $M_{1}=\begin{bmatrix}
1 & 0 & 0 \\
-4/3 & 1 & 0 \\
-1/3 & 0 & 1
\end{bmatrix} $ and $M_{2}=\begin{bmatrix}
1 & 0 & 0 \\
0 & ... | $\begin{bmatrix}
1 & 0 & 0 \\
-4/3 & 1 & 0 \\
-1 & 1/2 & 1
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 \\
4/3 & 1 & 0 \\
1 & -1/2 & 1
\end{bmatrix}=
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
2/3 & 0 & 1
\end{bmatrix}.$
Thereore
$\begin{bmatrix}
1 & 0 & 0 \\
-4/3 & 1 &... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What number do you remove from $1!2!\cdots 99!100!$ to get a perfect square?
The title says it all, there is a product, as shown above, one of the factorials must be removed and the product will make a perfect square. Which one?
For example, you could remove $54!$?
| First of all, you can write every $(n!)$ as $(n-1)!\cdot n$.
However, for now, just do this for all the odd values of $n$ and you get
$$(2!)(2!)\cdot 3 \cdot(4!)(4!)\cdot 5\cdots (98!)(98!)\cdot 99\cdot 100!$$
Now call $(2!)(4!)\cdots (98!)=C$ to simplify and your expression is equal to
$$C^2\cdot 3\cdot 5\cdot 7\cdots... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2333541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 4
} |
Derivative of $(2x-1)(x+3)^{\frac{1}{2}}$ Find the derivative of
$(2x-1)(x+3)^{\frac{1}{2}}$
My try -
$(2x-1)(\frac{1}{2} (x+3)^{\frac{-1}{2}} (x+0) + (x+3)^{\frac{1}{2}} (2)$
$ = (x+3)^{\frac{-1}{2}} (\frac{1}{2}(2x-1) + 2(x+3) $
$= \frac{3x+5.5}{2 (x+3)^{\frac{1}{2}}} $
My numerator is wrong and should be
$6x+... | There are a few issues here - some stray parentheses and an incorrectly written derivative which became corrected on the next step. I'd proceed like this:
\begin{align*}
\frac d{dx} (2x-1)(x+3)^{1/2} &= \left[\frac d{dx} (2x-1)\right] \cdot (x+3)^{1/2} + (2x-1) \cdot \frac d{dx} (x+3)^{1/2}\\[0.3cm]
&= 2 (x+3)^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2333747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The sum of consecutive integers is $50$. How many integers are there? I started off by calling the number of numbers in my list "$n$". Since the integers are consecutive, I had $x + (x+1) + (x+2)...$ and so on. And since there were "$n$" numbers in my list, the last integer had to be $(x+n)$. This is where I got stuck.... | If your $n$ is odd, then the middle number has to be $50/n$. The odd divisors of $50$ are $1$ and $5$, which gives us two solutions $50=50$ and $8+9+10+11+12=50$
If $n$ is even, then $50/n$ is the half-integer between the middle two numbers. So $n$ has to be an even divisor of $100$, but not a divisor of $50$, so $n=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2334607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 1
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Series $\sum_{n=1}^{\infty} H_n\left[\zeta(8)-\frac{1}{1^8}-\frac{1}{2^8}-\frac{1}{3^8}\cdots\frac{1}{n^8}\right]$ How can we find the sum of this hard series any hint please
My trial is to express zeta(8) as a series but I don't know how and what to do
$$\sum_{n=1}^{\infty} H_n\left[\zeta(8)-\frac{1}{1^8}-\frac{1}{2... | We shall begin by calculating the multivariate zeta value $\zeta(7,1)$ so we have it to hand later. Using "well known formula" and Euler reflection formula, we have
\begin{eqnarray*}
\zeta(8)&=&\zeta(7,1)+\zeta(6,2)+\zeta(5,3)+\zeta(4,4)+\zeta(3,5)+\zeta(2,6) \\
\zeta(2)\zeta(6)&=&\zeta(8)+\zeta(6,2)+\zeta(2,6) \\
\zet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2336289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find volume bounded by 3 equations using integration The prompt is to find the volume of the solid bounded by the equations $x^2 + y^2 -2y = 0$, $z = x^2 + y^2$ and $z \ge 0$
Plotting the equations, I get something like this
We get a paraboloid and a cylinder.
How to know if I have to use double or triple integration... | $\textbf{Third method}$: find the volume of the cylinder, where $0\leq z\leq 4$, and then subtract from it the volume of the cylinder whose lower bound is the paraboloid and upper bound is the plane $z=4$.
So rather than working with the equations:
$$
\color{blue}{z=0, \hspace{4mm} z=x^2+y^2, \hspace{4mm} \mbox{ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2336489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Suppose $f(x)$ is a real valued polynomial function of degree $6$ satisfying the following conditions
Suppose $f(x)$ is a real-valued polynomial function of degree $6$ satisfying the following conditions
(a) $f$ has minimum value at $x=0$ and $2$
(b) $f$ has maximum value at $x=1$
(c) $\displaystyle\lim_{x\to 0}\frac{... | Finding such a polynomial is not terribly difficult. Starting with
$$p(x)=ax^6+bx^5+cx^4+dx^3+ex^2+fx+g,$$
$p'(0)=0$ implies $f=0$, and in order for $\log(1+p(x)/x^3)/x$ to tend to $2$ when $x\to 0$, we need $\log(1+p(x)/x^3)$ to behave like $2x$ when $x\to 0$, so we could ask $p(x)/x^3$ to behave like $2x$ when $x\to ... | {
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"source": "stackexchange",
"question_score": "2",
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For how many integers $(3n)^4-(n-10)^4$ is a perfect square?
For how many integers $n$ the expression
$$(3n)^4-(n-10)^4$$
becomes a perfect square?
$$$$One way is: Let $x=3n$ and $y=n-10$ to get $$x^4-y^4=z^2.$$ This equation does not have non-trivial solutions in integers (I do not want to discuss about this).... | Using
$$\gcd(N+1,2N-1)=\gcd(N+1,-3)\mid 3$$
$$\gcd(N+1,5N^2-2N+2)=\gcd(N+1,-7N+2)=\gcd(N+1,9)\mid 9$$
$$\gcd(2N-1,5N^2-2N+2)=\gcd(2N-1,5)\mid 5 $$
we see that the factors are almost coprime, hence must almost (i.e., up to small factors involving at most a $2$, a $3$, and/or a $5$) be square.
Then from $\frac{2N-1}{N+1}... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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What is the limit of the following function? $\lim_{n\rightarrow \infty }\left ( n-1-2\left (\frac{\Gamma(n/2)}{\Gamma((n-1)/2)} \right )^2 \right )$
| Note that
$$
{{\Gamma (n/2)} \over {\Gamma (n/2 - 1/2)}} = \left( {n/2 - 1/2} \right)^{\,\overline {\,1/2\,} }
$$
where $x^{\,\overline {\,a\,}} $ denotes the Rising Factorial (rising Pochammer).
It is an increasing function for $1<n$.
Because of that and considering the rules for summing the exponents of the Rising F... | {
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"source": "stackexchange",
"question_score": "5",
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If $x$ is real, evaluate $k$ in absolute inequality
If $x$ is real and $$y=\frac{(x^2+1)}{x^2+x+1}$$ then it can be shown that $$\left|y-\frac{4}{3}\right|\leq k$$ Evaluate $k$
My attempt,
\begin{align}\left(y-\frac{4}{3}\right)^2&\leq k^2\\
\sqrt{\left(y-\frac{4}{3}\right)^2}&\leq k\end{align}
I don't know how to pr... | If $x=-1$ then $\left|\frac{x^2+1}{x^2+x+1}-\frac{4}{3}\right|=\frac{2}{3}$.
We'll prove that it's a maximal value.
Indeed, we need to prove that
$$\left|\frac{x^2+1}{x^2+x+1}-\frac{4}{3}\right|\leq\frac{2}{3}$$ or
$$-\frac{2}{3}\leq\frac{x^2+1}{x^2+x+1}-\frac{4}{3}\leq\frac{2}{3}$$ or
$$\frac{2}{3}\leq\frac{x^2+1}{x^2... | {
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"question_score": "2",
"answer_count": 7,
"answer_id": 5
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Find coefficient of $x^2$ in a complicated expansion
Find the coefficient of $x^2$ in the expansion of $(4-x^2)[(1+2x+3x^2)^6+(1+4x^3)^5]$
I noticed that since we only care about the coefficient of $x^2$, only the coefficients of $x^2$ inside the square brackets, as well as the constant terms, will matter. From there... | For a question like this cheating is legal :)
$\left(4-x^2\right) \left[\left(4 x^3+1\right)^5+\left(3 x^2+2 x+1\right)^6\right]=-1024 x^{17}+4096 x^{15}-2009 x^{14}-2916 x^{13}+1718 x^{12}+1844 x^{11}+15417 x^{10}+31144 x^9+33976 x^8+29832 x^7+20681 x^6+10488 x^5+4302 x^4+1428 x^3+310 x^2+48 x+8$
The answer is $310$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Baby Rudin: Theorem 3.20 (c) and (d) In theorem 3.20, Rudin offers a proof to the limits of the following sequences:
$$u_n = n^{\frac{1}{n}}$$
$$v_n = \frac{n^{\alpha }}{(1+p)^n}$$ with $\alpha$ a real number and $p > 0$.
In both the proofs, Rudin uses inequalities which I am not familiar with. For $u_n$, the following... | Let $f(x)=(1+x)^n-1-\frac{n(n-1)}{2}x^2$, where $x\geq0$.
For $n=1$ we see that $f(x)\geq0$.
But for $n\geq2$ we obtain: $f''(x)=n(n-1)(1+x)^{n-2}-n(n-1)\geq0$, which gives
$f'(x)=n(1+x)^{n-1}-n(n-1)x\geq f'(0)>0$ and
$f(x)\geq f(0)=0$ and we are done!
| {
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"url": "https://math.stackexchange.com/questions/2344430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 2
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Prove that $S_i<1$ for all $i$ by induction We define $x_1=\frac12$, and $x_n=(1-\frac{3}{2n})\cdot x_{n-1}$
Then we define $S_i=x_1+x_2+\cdots+x_i$
Prove that $S_i<1$ for all $i$
I can see that $x_n=(1-\frac{3}{2n})\cdot (1-\frac{3}{2n-2})\cdots (1-\frac 34)\cdot \frac12$
But what then? I suppose my observation isn't... | I think it will help to observe,
\begin{align*}
x_n &= \frac{1}{2} \cdot \frac{2n - 3}{2n} \cdot \frac{2n - 5}{2n - 2} \cdot \ldots \cdot \frac{1}{4} \\
&= \frac{(2n - 3)(2n - 5) \ldots 1}{(2n)(2n - 2)(2n - 4) \ldots 2} \\
&= \frac{1}{2^n n!}(2n - 3)(2n - 5) \ldots 1 \\
&= \frac{1}{2^n n!} \cdot \frac{(2n - 2)(2n - 3)(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2344814",
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"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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BMO2 1997 - Combinatorics Find the number of polynomials of degree 5 with distinct coefficients from the set {1, 2, 3, 4, 5, 6, 7, 8} that are divisible by $x^2-x+1$.
I tried multiplying $x^2-x+1$ by $ax^3+bx^2+cx+d$ to get $ax^5+(b-a)x^4+(a-b+c)x^3+(b-c+d)x^2+(c-d)x+d$, but I am struggling to count from here.
| Filling in the details for @Gerry Myerson's answer:
WLOG assume $d>a$ and $d < b < f$. We will count the number of pairs, and multiply by $3! \cdot 2 = 12$ to account for the symmetry. Do casework on the value of $d-a$.
If $d-a = 1$, then there are $10$ possible values for $(a,d,e,b,c,f)$:
$$(1,2,3,4,5,6), (1,2,3,4,6,... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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An unusual identity Prove that $$\left(\frac{\sqrt[3]{81+33\sqrt6}+\sqrt[3]{81-33\sqrt6}}{6}\right)^m=\left(\frac{\sqrt[5]{41+29\sqrt2}+\sqrt[5]{41-29\sqrt2}}{2}\right)^n$$ for all natural integers $m,n$.
| Hint: for the LHS let $\,a=\sqrt[3]{81+33\sqrt6}\,, \,b=\sqrt[3]{81-33\sqrt6}\,$ then $\,a^3+b^3=162\,$ and $\,ab=3\,$, so: $$\,(a+b)^3=a^3+b^3+3ab(a+b)=162+9(a+b)\,$$
But the equation $x^3-9x-162=0$ has only one real root which is $x=6$, therefore $a+b=6$.
[ EDIT ] As pointed out by @LordSharktheUnknown, the RHS woul... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Using cylindrical coordinates to evaluate the volume of a sphere external to an ellipsoid I need to use cylindrical coordinates to evaluate the volume of the sphere $$x^{2}+y^{2}+z^{2} = a^{2}$$ that's external to the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{a^2}+\frac{z^2}{b^2} = 1$.
So, $x$ and $y$ are equal for both, a... | What you need is to compute the following integral
$$\int\int\int_{\Delta V}r dr d\theta dz$$ in cylindrical coordinates, where $\Delta V$ is your volume of integration. The first thing I notice is that $b<a$, otherwise the sphere is inside the ellipsoid. So now all we need is to find the limits of this integration vol... | {
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Prove the inequality using AM-GM only. By considering "Arithmetic mean $\geq $ Geometric mean" prove the trigonometric inequality:
$$\sin A + \sin B + \sin C \leq \frac{3\sqrt{3}}{2}. $$
where $A+B+C=180°$.
My try: By using transformation formulae, I proved that
$$\sin A + \sin B + \sin C = 4\cos\left(\frac{A}{2}\rig... | Let $a$, $b$ and $c$ be sides-lengths of the triangle and $S$ be an area of the triangle.
We need to prove that
$$\sum_{cyc}\frac{2S}{bc}\leq\frac{3\sqrt3}{2}$$ or
$$4S(a+b+c)\leq3\sqrt3abc$$ or
$$(a+b+c)^3(a+b-c)(a+c-b)(b+c-a)\leq27a^2b^2c^2.$$
Now, let $a+b-c=z$, $a+c-b=y$ and $b+c-a=x$.
Hence, $x$, $y$ and $z$ are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2347366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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What is equivalent that logical function?
(A+B) => (B XOR C)
=> is Implication
Options:
BC+!A+!B+!C
!B!C+ABC
B!C+A+!BC
!BC+!ABC
BC+A!B!C
! is NOT
Please tell me how it is solved.
I do not think that this requires knowledge of discrete mathematics.
But I did not find any simple options
| $$(A+B) \Rightarrow (B \ XOR \ C) = \text{ (Rewrite XOR)}$$
$$(A+B) \Rightarrow (B!C + !BC) = \text{ (Rewrite} \Rightarrow )$$
$$!(A + B)+B!C + !BC = \text{ (DeMorgan)}$$
$$!A!B+B!C+!BC$$
.... which is not any of the options you are providing ...
$BC + !A +!B+!C=1$ for $A = 0$ and $B=C=1$, but then $!A!B+B!C+!BC = 0$
... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Other ways to evaluate the integral $\int_{-\infty}^{\infty} \frac{1}{1+x^{2}} \, dx$?
$$\int_{-\infty}^{\infty}\frac{1}{x^2+1}\,dx=\pi $$
I can do it with the substitution $x= \tan u$ or complex analysis. Are there any other ways to evaluate this?
| Derivation using series the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} = \frac{\pi}{4}$.
The integrand is an even function & break the interval at $1$ and we have
\begin{eqnarray*}
\int_{-\infty}^{\infty} \frac{dx}{1+x^2} =2 \int_{0}^{\infty} \frac{dx}{1+x^2} =2 \int_{0}^{1} \frac{dx}{1+x^2}+2 \underbrace{\int_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2349224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 9,
"answer_id": 8
} |
Finding the domain and the range of this function Problem: Find the domain and range of the function $f(x)=\log \sqrt{4-x^2}$
My attempt:
For domain
$y=f(x)=\log \sqrt{4-x^2}$
$\implies y=\log \sqrt{4-x^2}$
Since
logarithm of only positive numbers is possible,
$4-x^2>0$
$\implies x^2-4<0$
$\implies(x+2)(x-2)<0$
$\impl... | $\sqrt {4-x^2}$ cannot be greater than $\sqrt 4=2$, so $\log \sqrt{4-x^2}$ cannot be any greater than $\log \sqrt 4=\log 2$. It achieves that value when $x=0.$ As $\sqrt{4-x^2}$ can achieve $0$, you can have $\log \sqrt{4-x^2}$ as negative as you want, so the range is $(-\infty, \log 2]$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2352753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Find all solutions of $9^x-4^y=5^z$ over the naturals Find all solutions of $9^x-4^y=5^z$ over the naturals.
My attempt: We have
$${(3^x)}^2-{(2^y)}^2=5^z \Rightarrow (3^x+2^y)(3^x-2^y)=5^z.$$
It can be easily proved that $\gcd (3^x+2^y,3^x-2^y)=1$ so we have
$$3^x-2^y=1,3^x+2^y=5^z.$$
What do we have to do from here?
| As @ Xam said,
if $y=0$, then there is no solution,
if $y=1$, then $x=1$.
Now if $y$ is greater than $1$, then $2^y$ is divisible by $4$.
using modular arithmetic module $4$ we have:
$(-1)^x-0$ is congruent to $1$ module $4$, so $x$ must be even, i.e. $x=2k$.
Now $2^y=3^{2k}-1=(3^k-1)(3^k+1)$, and by the same conclus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2353835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Find a cubic polynomial. If $f(x)$ is a polynomial of degree three with leading coefficient $1$ such that $f(1)=1$, $f(2)=4$, $f(3)=9$, then $f(4)=?,\ f(6/5)=(6/5)^3?$
I attempt:
I managed to solve this by assuming polynomial to be of the form $f(x)=x^3+ax^2+bx+c$, then getting the value of $a,b,c$, back substituting i... | Classic long method:
Let $f(x) = x^3 + b x^2 + c x + d$ with $f(1) = 1$, $f(2) = 4$, $f(3) = 9$, which leads to
\begin{align}
f(1) &= 1 = 1 + b + c + d \hspace{10mm} \to d = -b - c \\
f(2) &= 4 = 8 + 4 b + 2 c + d = 8 + 3b + c \hspace{10mm} \to c = -4 - 3b, \, d = 4 + 2b \\
f(3) &= 9 = 27 + 9b + 3c + d = 19 + 2b
\end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2354401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find the distribution of $Y+X$ Consider the following two experiments: the first has outcome $X$ taking on
the values $0, 1$ and $2$ with equal probabilities; the second results in an (in-
dependent) outcome $Y$ taking on the value $3$ with probability $1/4$ and $4$ with
probability $3/4$. Find the distribution of $Y+X... | In this case you can enumerate all the possibilities and use independence to find their probabilities. Then you can divide up the possibilities according to what the sum $X+Y$ is.
We have:
$$
P(X=0,Y=3) = \frac{1}{3}\frac{1}{4} = \frac{1}{12}\\
P(X=1,Y=3) = \frac{1}{3}\frac{1}{4} = \frac{1}{12}\\
P(X=2,Y=3) = \frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2358120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Find minimum of $a+b+c+\frac1a+\frac1b+\frac1c$ given that: $a+b+c\le \frac32$
Find minimum of $a+b+c+\frac1a+\frac1b+\frac1c$ given that: $a+b+c\le \frac32$ ($a,b,c$ are positive real numbers).
There is a solution, which relies on guessing the minimum case happening at $a=b=c=\frac12$ and then applying AM-GM inequal... | WLOG $a\geq b\geq c.$
Let $f(a,b,c)=a+b+c+1/a+/b+1/c.$
For a given value of $a+b+c,$ let $c$ remain constant while $a,b$ vary , subject to the constraint that $a+b$ is constant, so that $a+b+c$ also remains constant. Then $db/da=-1$ and $d(1/b)/da=1/a^2 .$ So with constant $c$ we have $$df(a,b,c)/da=-1/a^2+1/b^2=(a-b)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2360813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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$x^7+y^7+z^7$ is divisible by $7^3$, then $x+y+z$ is divisible by $7^2$
Let $x, y, z$ be positive integers, and $7 \nmid xyz$. If $7^3|x^7+y^7+z^7$, show that $7^2|x+y+z$.
by Fermat's little theorem, $x^7 \equiv x \pmod7$, then $x^7+y^7+z^7\equiv x+y+z \equiv 0 $ (mod 7)
so we have $7 | (x+y+z)$.
what should I do nex... | Since
$$x^7+y^7+x^7=\sum_{cyc}(x^7-x)+x+y+z,$$
we see that $x+y+z$ is divided by $7$.
In another hand,
$$x^7+y^7+z^7=(x+y+z)^7-7(xy+xz+yz)(x+y+z)^5+7xyz(x+y+z)^4+$$
$$+14(xy+xz+yz)^2(x+y+z)^3-21xyz(xy+xz+yz)(x+y+z)^2-$$
$$-7(xy+xz+yz)^3(x+y+z)+7x^2y^2z^2(x+y+z)+7xyz(xy+xz+yz)^2,$$
which says (see the last term) that $(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2361499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
$\int (x^2+1)/(x^4+1)\ dx$ I have Divided the numerator and Denominator by $x^2$
to get $\dfrac{1+x^{-2}}{x^2+x^{-2}}$ then changed it into $(1+(x^{-2}))/[(x-x^{-1})^2 +2]$ then took $x-(1/x)$ as $u$ and Differentiated it with respect to $x$ to get $dx=du/(1+x^{-2})$ Finally I got this expression:
$$
\int\frac{x^2+1}{... | Standard tables of integrals say:
$$ \int (1+v^2)^{-1} \, dv = (\arctan v) + \text{constant}. $$
Therefore
\begin{align}
\int (2+u^2)^{-1} \, du & = \frac 1 {2\sqrt 2} \int \left(1 + \left(\frac u {\sqrt 2} \right)^2 \right)^{-1} \, \big( \sqrt 2\, du\big) \\[10pt]
& = \frac 1 {2\sqrt 2} \int (1+v^2)^{-1} \, dv \\[10pt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2361853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 0
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$\begin{vmatrix} 1 & a &bc \\ 1& b & ac\\ 1&c & ab \end{vmatrix}=\begin{vmatrix} 1 & a &a^2 \\ 1& b&b^2 \\ 1& b & c^2 \end{vmatrix}$
Prove that \begin{align}\begin{vmatrix}
1 & a &bc \\
1& b & ac\\
1&c & ab
\end{vmatrix}&=\begin{vmatrix}
1 & a &a^2 \\
1& b&b^2 \\
1& b & c^2
\end{vmatrix}\\&=(c-a)(b-a)(c-b)\... | \begin{align}\begin{vmatrix}
1 & a &bc \\
1& b & ac\\
1&c & ab
\end{vmatrix}&=\begin{vmatrix}
1 & a &a^2 \\
1& b&b^2 \\
1& c & c^2
\end{vmatrix}\\\\
\begin{vmatrix}
1 & a &bc \\
0& b-a & ac-bc\\
0&c-a & ab-bc
\end{vmatrix}&=\begin{vmatrix}
1 & a &a^2 \\
0& b-a&b^2-a^2 \\
0& c-a & c^2-a^2
\end{vmatri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2362387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Rotate the parabola $y=x^2$ clockwise $45^\circ$. I used the rotation matrix to do this and I ended up with the equation: $$x^2+y^2+2xy+\sqrt{2}x-\sqrt{2}y=0$$
I tried to plot this but none of the graphing softwares that I use would allow it.
Is the above the correct equation for a parabola with vertex (0,0) and axis o... | if you want to prove that the axis of symmetry rotates with the function then simply show that $M(\theta)R(\theta)(x,y)=R(\theta)(x',y')$ where $R$ and $M$ are rotation and reflection matrices respectively given that $\theta=0$ is the line of symmetry.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 3
} |
Find all possible values of $x+y+z$. Let $x, y, z$ be non-zero integers such that ${x\over y}+{y\over z}+{z\over x}$ is an integer. Find all possible values of $x+y+z$. Please provide a proof with all solutions.
| Note: this assumes that the question is really about non-zero integers, and not about positive integers. I think the answer is more difficult when having to deal with positive integers.
Now, we can get any non-zero integer $t$ by choosing $(x,y,z)=(t,t,-t)$.
This gives $\frac{t}{t} + \frac{t}{-t} + \frac{-t}{t} = -1$ ... | {
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"url": "https://math.stackexchange.com/questions/2364572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Find n so that the following converges $\int_1^{+ \infty} \left( \frac{nx^2}{x^3 + 1} - \frac 1 {13x + 1} \right) dx$ Question
Determine $n$ such that the following improper integral is convergent
$$ \int_1^{+ \infty} \left(
\frac{nx^2}{x^3 + 1}
-\frac{1}{13x + 1}
\right) dx
$$
I'm not sure how to go abou... | $\int_1^{+ \infty} \left(
\frac{nx^2}{x^3 + 1}
-\frac{1}{13x + 1}
\right) dx
$
$\frac{nx^2}{x^3 + 1}
-\frac{1}{13x + 1}
=\frac{nx^2(13x+1)-(x^3 + 1)}{(x^3 + 1)(13x + 1)}
=\frac{x^3(13n-1)+nx^2-1}{(x^3 + 1)(13x + 1)}
$.
If $13n-1 \ne 0$,
this behaves like
$\frac1{x}$
and the integral diverges.
Therefo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2365228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
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Modulus of complex numbers problem draw the loci Given $|z − 3| + |z + 3| = 12$, $z = x + iy$, find the expression in terms of $x$ and $y$ and draw the loci.
I've tried many ways and I cannot isolate $x$ or $y$. I think that is necessary put the equation in this form $(x-a)^2 + (y-b)^2 = r^2$.
May someone help me?
| If $z=x+iy$ where $x,y$ are real
For
$$\sqrt{(x-c)^2+y^2}+\sqrt{(x+c)^2+y^2}=2a$$
$$(x+c)^2+y^2=4a^2+(x-c)^2+y^2-4a\sqrt{(x-c)^2+y^2}$$
$$\iff a\sqrt{(x-c)^2+y^2}=a^2-cx$$
Squaring we get $$a^2\{(x-c)^2+y^2\}=a^4-2a^2cx+c^2x^2$$
$$\iff(a^2-c^2)x^2+a^2y^2=a^2(a^2-c^2)$$
For $a\ne c,$ $$\dfrac{x^2}{a^2}+\dfrac{y^2}{a^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2367608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Explain why the graph of $y=\frac{4x}{x^2+1}$ and $y=2\sin(2\arctan x)$ are the same. Explain why the graph of $y=\frac{4x}{x^2+1}$ and $y=2\sin(2\arctan x)$ are the same.
The first equation is of the form of Newton's Serpentine. When you graph the second equation it appears to overlap the first equation.
I'm not sure ... | Since $\sin(2A) = 2 \sin A \cos A$,
\begin{align}2\sin(2 \arctan x) &= 4 \sin (\arctan x) \cos(\arctan x) \\
&=4\left( \frac{x}{\sqrt{1+x^2}} \right)\left( \frac{1}{\sqrt{1+x^2}} \right) \\
&=\frac{4x}{1+x^2}\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2367735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
$\cos^2 76^{\circ} + \cos^2 16^{\circ} -\cos 76^{\circ} \cos 16^{\circ}$
Find the value of $$\cos^2 76^{\circ} + \cos^2 16^{\circ} -\cos 76^{\circ} \cos 16^{\circ}$$
I did it like this
$$\cos^2 76^{\circ}+\cos^2 16^{\circ} = \cos(76^{\circ}+16^{\circ}) \, \cos(76^{\circ}-16^{\circ}).$$
So the expression is $$\cos 92^... | Let $c= \cos 16^\circ$ and $s=\sin 16^\circ$. Cosine addition gives ($76^\circ=60^\circ+16^\circ$)
\begin{eqnarray*}
\cos(76^\circ)=\frac{c-\sqrt{3}s}{2}
\end{eqnarray*}
So the expression is
\begin{eqnarray*}
\cos^2(76^\circ)+\cos^2(16^\circ)-\cos(76^\circ)\cos(16^\circ) =\left(\frac{c-\sqrt{3}s}{2}\right)^2+c^2-c\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2368243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Numbers of ways to choose $k$ out of the first $n$ natural numbers so the longest string of consecutive numbers is exactly $m$? How many ways can $k$ numbers be chosen from the first $n$ natural numbers so that the longest string of consecutive numbers is exactly $m$ numbers long
For example, if choosing $k = 7$ distin... | Another more practical solution uses a recurrence relation. Let the size of the gaps between the $n-k$ numbers not chosen be $x_1,x_2,...,x_{n-k+1}$. Then the problem reduces to how many ways are there such that $\forall i$, $x_i\in [0,1...m]$, $\sum_{i=1}^{n-k+1}x_i = k$ and at least one of $x_i = m$.
To solve this f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2368624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Complex Integration using Cauchy's Theorem The problem is the integration of
$$I=\int_{\left\lvert z-1\right\rvert=1} f(z) dz$$
where
$$f(z)=\frac{1}{z^3-1}$$
and the path goes $1$ loop in positive direction.
I tried to solve the problem using Cauchy's Theorem by finding $z$ that makes $f(z)$ denominator be $0$. That... | Note: The hint of @rtybase is essentially the answer to OPs question. Let's recall Cauchy's Integralformula:
If $a$ is in the interior of $\gamma=\{z:|z-1|=1\}$ and a function $g$ is holomorphic in a region which contains the closure of the interior of $\gamma$, then
\begin{align*}
g(a)=\frac{1}{2\pi i}\oint_{\gamma}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2372902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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If $ a^{2} + b^{2} = c^{2}$ and $c$ is even, prove that $a$ and $b$ are both even. Question:
If $ a^{2} + b^{2} = c^{2}$ and $c$ is even, prove that $a$ and $b$ are both even.
I am not quite sure how to prove this. My guess is proof by contradiction.
Assume the contrary, that is, $a^{2} + b^{2} = c^{2}$ for $c$ even an... | Suppose $a^2+b^2=c^2$ and $c$ is even. Since $a^2=c^2-b^2=(c+b)(c-b)$, $a$ and $b$ have the same parity. Assume $a$ is odd. Then $a^2+b^2\equiv 2(\mod4)$, whence $c^2\equiv2(\mod 4)$ which is a contradiction since $4\mid c^2$ as $2\mid c$. Hence $a$ and $b$ are even.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2375495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Find the area bounded by $ \ x^4+y^4=4(x^2+y^2) \ $ . Find the area bounded by $ \ x^4+y^4=4(x^2+y^2) \ $ .
Answer:
The graph is above :
Since the region is symmetrical ,
$ Area =4 \times \int_{0}^{2} \int_{0}^{\sqrt{2+\sqrt{4x^2-x^4+4}}} dxdy $
Am I right ? Is there any help ?
| Convert to polar form $r^2=\dfrac{16}{\cos 4 t+3}$
The area is $$A=\frac{1}{2} \int_0^{2 \pi } \frac{16}{\cos 4 t+3} \, dt=32 \int_0^{\frac{\pi }{2}} \frac{1}{\cos 4 t+3} \, dt$$
$\cos 4t=\cos^2 2t-\sin^2 2t=2\cos^2 2t -1$
so we have to integrate
$\int \dfrac{dt}{2(\cos^2 2 t+1)}=\dfrac{1}{2}\int \dfrac{dt}{1+\cos^2 2t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2377389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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How to simplify an expression involving several square roots without a calculator? $$\frac{5 \sqrt{7}}{4\sqrt{3\sqrt{5}}-4\sqrt{2\sqrt{5}}}- \frac{4 \sqrt{5}}{\sqrt{3\sqrt{5}}-\sqrt{2\sqrt{5}}}$$
This type of questions are common in the university entrance examinations in our country but the calculators are not allowed... |
Let $\sqrt{a\sqrt{b}} = \sqrt{a}\sqrt{\sqrt{b}} = \sqrt{a}\cdot\sqrt[4]{b}$
$$\tag1
\frac{5 \sqrt{7}}{4\sqrt{3\sqrt{5}}-4\sqrt{2\sqrt{5}}}- \frac{4 \sqrt{5}}{\sqrt{3\sqrt{5}}-\sqrt{2\sqrt{5}}}$$
$$\tag2
\frac{5\sqrt{7}}{4\sqrt{3}\sqrt[4]{5}-4\sqrt{2}\sqrt[4]{5}}- \frac{4\sqrt{5}}{\sqrt{3}\sqrt[4]{5}-\sqrt{2}\sqrt[4]{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2377485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Infinite series inequality. I have some trouble with the following problem:
Show that $$\sum_{n=1}^{\infty} \frac{1}{(n+1)\sqrt[p]{n}}<p$$ is true for every natural number $p$.
I tried using induction but I can't finish the proof. Or there is a simpler way to solve this? Thank you.
| Solution 1. Let $a = \sqrt[p]{n}$ and $b = \sqrt[p]{n+1}$. If $p > 1$ is an integer, then
\begin{align*}
\frac{1}{(n+1)\sqrt[p]{n}}
&= \frac{p}{ab \cdot pb^{p-1}} \\
&< \frac{p}{ab \cdot (b^{p-1} + ab^{p-2} + \cdots + a^{p-1})} \\
&= p \cdot \frac{b - a}{ab (b^p - a^p)} \\
&= p \left( \frac{1}{\sqrt[p]{n}} - \frac{1}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2378857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to turn the ellipse $x^2 - xy + y^2 - 3y - 1 = 0$ to the canonical form using an isometric transformation? There is an exam problem I'm having trouble with, it is as follows:
Turn the equation $x^2 - xy + y^2 - 3y -1 = 0$ into the canonical form using an isometric transformation and write down the transformation.
... | To keep it understandable albeit inelegant I'll do the passage in two steps
First the translation
$x^2 - xy + y^2 - 3y -1 = 0$
we look for a new centre $(h;\;k)$ so we substitute $x=x'+h;\;y=y'+k$
$-(h+x) (k+y)+(h+x)^2+(k+y)^2-3 (k+y)-1=0$
$x'^2-x' y'+y^2+x' (2 h-k)+y' (-h+2 k-3) +h^2-h k+k^2-3 k-1=0$
To have no first... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2381286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
What are the roots of this equation? I have a quadratic equation $ ax^2 +bx+c =0 $, where $ a,b,c $ all are positives and are in Arithmetic Progression.
Also, the roots $\alpha$ and $\beta$ are integers.
I need to find out $ \alpha + \beta + \alpha\beta $.
I have tried taking $ a = a' - d , b = a' , c = a' + d $ becau... | If the roots are integers, we can write $x^2 + bx + c = (x-\alpha)(x-\beta)$. Note that it's okay to take $a=1$, since $b-a = c-b \implies \frac{b-a}{a} = \frac{c-b}{a}$, i.e. the arithmetic progression is preserved.
Now write $c = b + (b-1) = 2b-1$.
Use Vieta's: $\alpha + \beta = -b$ and $\alpha\beta = 2b-1$.
$\alpha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2381507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find the value of $\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ca}+\frac{c^2}{2c^2+ab}$
Let $a$, $b$ and $c$ be real numbers such that $a + b + c = 0$ and define:
$$P=\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ca}+\frac{c^2}{2c^2+ab}.$$
What is the value of $P$?
This question came in the regional maths olympiad. I tried AM-GM a... | For $\prod\limits_{cyc}(a-b)\neq0$ we obtain:
$$\sum_{cyc}\frac{a^2}{2a^2+bc}=\sum_{cyc}\frac{a^2}{a(a+b+c)+a^2-ab-ac+bc}=$$
$$=\sum_{cyc}\frac{a^2}{(a-b)(a-c)}=\frac{\sum\limits_{cyc}a^2(c-b)}{\prod\limits_{cyc}(a-b)}=1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2384363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Find all solutions to $x^3+(x+1)^3+ \dots + (x+15)^3=y^3$
Find all pairs of integers $(x, y)$ such that
$$x^3+(x+1)^3+ \dots + (x+15)^3=y^3$$
What I have tried so far:
The coefficient of $x^3$ is $16$ in the left hand side. It is not useful then to trying bound LHS between, for example, $(ax+b)^3$ and $(ax+c)^3$ a... | This isn't a complete solution, but I hope it gives you an approach. (It's too long for a comment)
Since we have $$\sum_{r=1}^{n} r^3=\left(\frac { n(n+1)}{2}\right)^2$$
You can write
\begin{align}
x^3+(x+1)^3+ \dots + (x+15)^3
&=\sum_{r=1}^{x+15} r^3-\sum_{r=1}^{x-1} r^3
\\
&=\left(\frac { (x+15)(x+16)}{2}\right)^2-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2385215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 0
} |
Finding third row of orthogonal matrix?
Find a $3\times3$ orthogonal matrix whose first two rows are $\Big[\frac{1}{3},\frac{2}{3},\frac{2}{3}\Big]$ and $\left[0,\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right]$.
I tried two approaches.
One, finding vector cross product of given two rows.
Second, assuming third row as $... | One of the possible solutions:
$$\frac{\begin{pmatrix} \frac{1}{3} \\\frac{2}{3} \\ \frac{2}{3}\end{pmatrix} \times \begin{pmatrix} 0 \\\frac{1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}}\end{pmatrix}}{\left\Vert\begin{pmatrix} \frac{1}{3} \\\frac{2}{3} \\ \frac{2}{3}\end{pmatrix} \times \begin{pmatrix} 0 \\\frac{1}{\sqrt{2}} \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2385989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Find the real and imaginary part of z let $z=$ $$ \left( \frac{1 + \sin\theta + i\cos\theta}{1 + \sin\theta - i\cos\theta} \right)^n$$
Rationalizing the denominator:
$$\frac{1 + \sin\theta + i\cos\theta}{1 + \sin\theta - i\cos\theta}\cdot\left( \frac{1 + \sin\theta + i\cos\theta}{1 + \sin\theta + i\cos\theta}\right) =... | Following your rationalizing the denominator method, you err when expanding the square. You should get
$$
\frac{1 + \sin\theta + i\cos\theta}{1 + \sin\theta - i\cos\theta} = \frac{(1+\sin\theta)^2 - (\cos\theta)^2 + 2i\cos\theta(1+\sin\theta)}{(1+\sin\theta)^2 + (\cos\theta)^2}
$$
This can then be simplified by expandi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2387172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 8,
"answer_id": 7
} |
Finding $n$th power of a $3\times 3$ matrix Find the $A^n$ if $$A=\begin{bmatrix}1 & a & b \\0 & 1 &a\\0 &0 &1\end{bmatrix}$$
I tried inductive method to show $$A^n=\begin{bmatrix}1 & na & nb+\frac{n(n-1)}{2}a^2 \\0 & 1 &na\\0 &0 &1\end{bmatrix}$$
now : My question is : Is there other method (idea ) to find $A^n$ ?
T... | $$A=\begin{bmatrix}1 & a & b \\0 & 1 &c\\0 &0 &1\end{bmatrix}$$
Write $A = I + U$, where $U^2 = \begin{bmatrix} a \\ 0 \\ 0 \end{bmatrix} \begin{bmatrix} 0 & 0 & c \end{bmatrix}$ and $U^3=0$.
$$ A^n=(I+U)^n=I+nU+{n\choose 2}U^2 =
\pmatrix{1&na&nb+\frac{n(n-1)}{2}ac\\0&1&nc\\0&0&1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2388294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Solving $\frac{1}{(x-1)} - \frac{1}{(x-2)} = \frac{1}{(x-3)} - \frac{1}{(x-4)}$. Why is my solution wrong? I'm following all hitherto me known rules for solving equations, but the result is wrong. Please explain why my approach is not correct.
We want to solve:
$$\frac{1}{(x-1)} - \frac{1}{(x-2)} = \frac{1}{(x-3)} - \f... | To avoid errors or pitfalls it can sometimes be helpful to try an initial 'lazy approach', where you still don't drop mathematical circumspection.
When looking at the equation a third degree polynomial can be sought, but that is no fun and we are really hoping that a quadratic equation 'pops-up'. The equation is ugly ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2388701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 5
} |
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.