Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
How to prove that $(-18+\sqrt{325})^{\frac{1}{3}}+(-18-\sqrt{325})^{\frac{1}{3}} = 3$ How to prove that $\left(-18+\sqrt{325}\right)^{\frac{1}{3}}+\left(-18-\sqrt{325}\right)^{\frac{1}{3}} = 3$ in a direct way ?
I have found one indirect way to do so: Define $t=\left(-18+\sqrt{325}\right)^{\frac{1}{3}}+\left(-18-\sqrt{... | By direct calculation we have the identity
$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ac).$
Then set $a=\sqrt[3]{-18+\sqrt{325}},b=\sqrt[3]{-18-\sqrt{325}}$ and $c=3$. Note that $ab=-1$ and $a^3+b^3=-36$, so $a^3+b^3+c^3=3abc$.
Since $(a-b)^2+(b-c)^2+(b-c)^2>0\Longleftrightarrow a^2+b^2+c^2-ab-bc-ac>0$ holds for any... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3011230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Limit of $\lim_{x \to 0} (\cot (2x)\cot (\frac{\pi }{2}-x))$ (No L'Hôpital)
$\lim_{x \to 0} (\cot (2x)\cot (\frac{\pi }{2}-x))$
I can't get to the end of this limit. Here is what I worked out:
\begin{align*}
& \lim_{x \to 0} \frac{\cos 2x}{\sin 2x}\cdot\frac{\cos(\frac{\pi }{2}-x )}{\sin(\frac{\pi }{2}-x )}
\lim_{x \... | Hint: $$\lim_{x \to 0} (\cot (2x)\cot (\frac{\pi }{2}-x))=\lim_{x \to 0} {\cos (2x)\over \sin 2x}\tan (x) =\lim_{x \to 0} {\cos (2x)\over 2\sin x\cos x}{\sin x\over \cos x}$$
$$=\lim_{x \to 0} {\cos (2x)\over 2\cos^2 x} = {1\over 2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3011543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 1
} |
Is it true that there aren't any three different numbers $x,y,z$ such that $x^3+x \equiv y^3+y \equiv z^3+z \pmod p $? Let $p$ be a prime number. Is it true that there aren't any three different numbers $x,y,z$ such that $$x^3+x \equiv y^3+y \equiv z^3+z \pmod p $$
with $x -y, y-z, z-x$, each of them cannot be divided... | Solutions exist for all primes $p\ge5,p\neq7$.
As the OP observed, we have the Vieta relation $x+y+z=0$ as $x,y,z$ are the zeros of the cubic
$$
P(T)=T^3+T+c=(T-x)(T-y)(T-z)
$$
in the field $\Bbb{F}_p$. Here $-c=-xyz$ is the shared value of $x^3+x,y^3+y$ and $z^3+z$ (treated as elements of $\Bbb{F}_p$ turning congruenc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3011725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Prove the sequence $a_n =\frac{1\cdot 3\cdot…\cdot(2n-1)}{2\cdot4\cdot…\cdot2n}$ has a limit I have several questions to ask:
1) Show increasing, find the upper bound if you can of
$\sqrt{(n^2-1)}/n$.
$\sqrt{(n^2-1)}/n= |n|\sqrt{1-1/n^2}/n$ if $n$ is positive than $\sqrt1$ else $-\sqrt1$;
bound: $\sqrt{(n^2-1)}/n \le... | $$\frac{(2n-1)!!}{(2n)!!}=\frac{1}{4^n}\binom{2n}{n}=\frac{2}{\pi}\int_{0}^{\pi/2}\left(\cos\theta\right)^{2n}\,d\theta $$
is quite clearly decreasing and convergent to zero (by the dominated convergence theorem).
Laplace's method gives $\frac{1}{4^n}\binom{2n}{n}\sim\frac{1}{\sqrt{\pi\left(n+\frac{1}{4}\right)}}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3012151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Show $11^{11}+12^{12}+13^{13} =10k$ without direct calculation Prove that $11^{11}+12^{12}+13^{13}$ is divisible by $10$.
Obviously you could just put that in to a calculator and see the results, but I was wondering about some of the other approaches to this? I have not studied modulus', so if you could explain it with... | The first digit of $11^{11}=$ the first digit of $1^{11}=$ $1$.
The first digit of $12^5$ equals the first digit of $2^5=$ the first digit of $32=2$.
$\text{(The first digit of $12^{10}) =$ (The first digit of $12^5)^2 = 4$ }$.
$\text{(The first digit of $12^{12}) =
$ (The first digit of $12^{10}) \times($The f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3013212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 11,
"answer_id": 7
} |
Solving $\int \frac{4x^2-3x+2}{4x^2-4x+3}$ with partial fractions I'm trying to understand how my textbook solved this problem but something seems a bit off.
First thing to do was to perform long division since the degree of the numerator is not less than the degree of the denominator, such a division yielded:
$$1 + \f... | HINT
We have that
$$\frac{4x^2-3x+2}{4x^2-4x+3}=\frac{4x^2-4x+3+x-1}{4x^2-4x+3}=1+\frac{x-1}{4x^2-4x+3}=$$
$$=1+\frac18\frac{8x-4-4}{4x^2-4x+3}=1+\frac18\frac{8x-4}{4x^2-4x+3}-\frac12\frac{1}{4x^2-4x+3}$$
$$=1+\frac18\frac{8x-4-4}{4x^2-4x+3}=1+\frac18\frac{8x-4}{4x^2-4x+3}-\frac12\frac{1}{2+(2x-1)^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3013535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Show that $\sqrt 5$ can be expressed as a polynomial in $e^{2\pi i/5}$ over $\Bbb Z$ Question from a Qualifying Exam:
*
*Show that $\sqrt 5$ can be expressed as a polynomial in $e^{(\frac{2\pi i}{5})}$ over $\Bbb Z$
*If in a field the equation $x^2-5$ has no solution then $x^5-1$ also has no non-trivial solution.
... | To your first question: Here is a high-faluting answer. If $p$ is any odd
prime number (i.e., any prime number $>2$), then the Gauss
sum is defined to be the
number
\begin{equation}
g\left( 1;p\right) :=\sum_{n=0}^{p-1}e^{2\pi in^{2}/p}.
\end{equation}
Gauss proved that
\begin{equation}
g\left( 1;p\right) =
\begin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3013766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Matrix equation where some entries are solution to a polynome
Let $z$ be a solution to $z^2+z+1=0$. Find a solution to
\begin{bmatrix}1&1&1&3\\1&1&1&-1\\1&z&z^2&0\\1&z^2&z&0\end
{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end
{bmatrix} =\begin{bmatrix}9\\1\\0\\0\end{bmatrix}
This an old exam question in my linear al... | $$\begin{bmatrix}1&1&1&3\\1&1&1&-1\\1&z&z^2&0\\1&z^2&z&0\end
{bmatrix}\begin{bmatrix}a\\a\\a\\b\end
{bmatrix} =\begin{bmatrix}3a+3b\\3a-b\\a(1+z+z^2)\\a(1+z+z^2)\end{bmatrix}=\begin{bmatrix}3a+3b\\3a-b\\0\\0\end{bmatrix}$$
We just have to solve for $3a+3b=9$ and $3a-b=1$.
We are already told that $1+z+z^2=0$, to use th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3022446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Is $h:x\to \exp\left[-\frac{x^2}{2}\right]\left(\exp\left[\frac{x^4}{24n}\right]-1\right)$ increasing? I struggling to show this function ins increasing on $\mathbb{R}^+$
$$h:x\to \exp\left[-\dfrac{x^2}{2}\right]\left(\exp\left[\dfrac{x^4}{24n}\right]-1\right) \qquad n\in \mathbb{N}^*$$
With the derivative
I've found
... | We have that
$$h'(x)=-xe^{-\frac{x^2}{2}}\left(e^{\frac{x^4}{24n}}-1\right)+\frac{x^3}{6n}e^{-\frac{x^2}{2}}e^{\frac{x^4}{24n}}
=-xe^{-\frac{x^2}{2}}\left(\left(1-\frac{x^2}{6n}\right)e^{\frac{x^4}{24n}}-1\right)$$
then we need to prove that
$$\left(\frac{x^2}{6n}-1\right)e^{\frac{x^4}{24n}}+1 \ge 0 $$
and for $\frac{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3022579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove a geometric sequence a, b, c from the arithmetic progression $1/(b-a)$, $1/2b$, $1/(b-c)$ The given task is:
The following forms an arithmetic sequence: $$\frac{1}{b-a}, \frac{1}{2b}, \frac{1}{b-c}.$$
Show, that $a, b, c$ forms an geometric sequence.
It's easily enough to understand that $$ \frac{1}{2b}-\frac{1}{... | We have
$$
\frac1b = \frac{1}{b-a} + \frac1{b-c}\\
(b-a)(b-c) = (b-c + b-a)b\\
b^2 - ab - bc + ac = 2b^2-bc - ab\\
ac = b^2
$$
and we are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3024160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that $({a\over a+b})^3+({b\over b+c})^3+ ({c\over c+a})^3\geq {3\over 8}$
Let $a,b,c$ be positive real numbers. Prove that $$\Big({a\over a+b}\Big)^3+\Big({b\over b+c}\Big)^3+ \Big({c\over c+a}\Big)^3\geq {3\over 8}$$
If we put $x=b/a$, $y= c/b$ and $z=a/c$ we get $xyz=1$ and
$$\Big({1\over 1+x}\Big)^3+\Big({1\... | By Holder $$\left(\sum_{cyc}\frac{a^3}{(a+b)^3}\right)^2\sum_{cyc}1\geq\left(\sum_{cyc}\sqrt[3]{\left(\frac{a^3}{(a+b)^3}\right)^2\cdot1}\right)^3=\left(\sum_{cyc}\frac{a^2}{(a+b)^2}\right)^3.$$
Thus, it's enough to prove that
$$\frac{\left(\sum\limits_{cyc}\frac{a^2}{(a+b)^2}\right)^3}{3}\geq\frac{9}{64}$$ or
$$\sum\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3025819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 2
} |
Textbook Proposition on Product of Real Analytic Functions
Let
\begin{align*}
\sum\limits_{j=0}^\infty a_j (x-c)^j && \sum\limits_{j=0}^\infty b_j (x-c)^j \\
\end{align*}
be two power series with intervals of convergence $\mathcal{C}_1$ and $\mathcal{C}_2$ centered on at $c$. Let $f_1(x)$ be the function defined b... | You can actually prove this by induction
Works for $N = 1$
\begin{eqnarray}
D_1 &=& a_0b_0 + (a_0b_1 + a_1 b_0)(x - c) \\ &=& a_0 [b_0 + b_1(x - c)] + a_1[b_0(x - c)] \\
&=& a_0 B_1 + a_1 B_0
\end{eqnarray}
Assume that it works for $N - 1$
$$
\sum_{m = 0}^{N - 1}\sum_{j + k = m}a_j b_k(x - c)^m = a_0 B_{N - 1}
+ a_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3027214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
value of $k$ in binomial expression If $\displaystyle \binom{404}{4}-\binom{4}{1}\cdot \binom{303}{4}+\binom{4}{2}\cdot \binom{202}{4}-\binom{4}{3}\cdot \binom{101}{4}=(101)^k.$ Then $k$ is
Iam trying to simplify it
$\displaystyle \frac{(404)!}{4!\cdot (400)!} -4\cdot \frac{(303)!}{4!\cdot (299)!}+6\cdot \frac{(202)!}... | Keep on simplifying:
$$\displaystyle \frac{(404)!}{4!\cdot (400)!} -4\cdot \frac{(\color{red}{303})!}{4!\cdot (299)!}+6\cdot \frac{(202)!}{(198)!\cdot 4!}-4\cdot \frac{(101)!}{4!\cdot (97)!}=\\
\displaystyle \frac{404\cdot 403\cdot 402\cdot 401}{24} - \frac{303\cdot 302\cdot 301\cdot 300}{6}+\frac{202\cdot 201\cdot 200... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3028138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Easier way to find eigenvalues of Matrices? I am trying to find eigenvalues for this matrix,
A =
$\begin{bmatrix}
3 & 2 & -3 \\
-3 & -4 & 9 \\
-1 & -2 & 5 \\
\end{bmatrix}$
I find the characteristic equation here:
$(\lambda I - A)
=
\begin{bmatrix}
\lambda - 3 & -2 & 3 \\
3 & \lambda + 4 & -9 \\
1 & 2 & \lambda - 5 \... | For 2 by 2 and 3 by 3 there are worthwhile recipes for the characteristic polynomial. For 2 by 2, it is $\lambda^2 - T \lambda + D,$ where $T$ is the trace and $D$ is the determinant.
more in a few minutes.
For 3 by 3, it is $$ \lambda^3 - \sigma_1 \lambda^2 + \sigma_2 \lambda - \sigma_3 $$
Here $\sigma_1$ is the trace... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3028870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How can we find minimum radius of circle which contains $\arctan^2(x)+\arctan^2(y)=a$? How can we find minimum radius of circle which contains $\arctan^2(x)+\arctan^2(y)=a$, where I think $a<\pi/2$?
For example I have some plots from WolframAlpha and I see it depends on $a$.
But have no idea how to find maximum radius... | Seems like the smallest circle is always tangent at $x=0$ or $y=0$. So you can put $\arctan(x)^2=a$ and solve $x=\tan\sqrt a$. This would be the radius.
Edit: let's make it rigorous.
Let's maximize $x^2+y^2$ with the constraint $\arctan(x)^2+\arctan(y)^2=a$. By symmetry we can work in the positive quadrant. With Lagran... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3030062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Find $f,g$ s.t. $f\circ g=\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\ 10 & 4 & 5 & 7 & 8 & 9 & 2 & 6 & 3 & 1\end{pmatrix}.$
Let $f$ and $g$ be permutations such that
$$f \circ f = id,$$
$$g \circ g = id,$$
and
$$f\circ g =\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
10 & 4 & 5 & 7 & 8 & 9 & 2 ... | A permutation $f$ is an involution if $f\circ f=id$.
As you know, any permutation can be written as a product of disjoint cycles; your permutation is $(1\ 10)(2\ 4\ 7)(3\ 5\ 8\ 6\ 9)$. In order to write an arbitrary permutation as a product of two involutions, it suffices (since disjoint permutations commute) to write ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3033171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Evaluating $1-\frac12-\frac13+\frac14+\frac15+\frac16-\cdots$
What is the value of $$S=1-\frac12-\frac13+\frac14+\frac15+\frac16-\cdots$$ where the sign alternates over the triangular numbers?
To start, it is easy to prove convergence. The sum of each set of reciprocals (e.g. $\{1/2,1/3\}$ and $\{1/4,1/5,1/6\}$) can ... | In other terms we want to evaluate
$$ \sum_{n\geq 1}(-1)^{n+1}\left(H_{n(n+1)/2}-H_{n(n-1)/2}\right)=\int_{0}^{1}\sum_{n\geq 1}(-1)^{n+1}\frac{x^{n(n+1)/2}-x^{n(n-1)/2}}{x-1}\,dx$$
where the theory of modular forms ensures
$$ \sum_{n\geq 0} x^{n(n+1)/2} = \prod_{n\geq 1}\frac{(1-x^{2n})^2}{(1-x^n)}=\prod_{n\geq 1}\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3034436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 1,
"answer_id": 0
} |
If $\frac {a}{3^{x-1}}=\frac{b}{3^{y+2}}=\frac{c}{3^{z-1}}=\frac 15\;$ then which of the following equals $a×b×c$? The problem is:
If $\frac {a}{3^{x-1}}=\frac{b}{3^{y+2}}=\frac{c}{3^{z-1}}=\frac 15,\;$ then which of the following equals $a×b×c$ ?
A) $\frac {1}{375}$
B) $\frac{1}{125}$
C) $\frac{27}{125}$
D) $\fra... | Yes some information is missing, indeed we have that
$$\frac {a}{3^{x-1}}=\frac{b}{3^{y+2}}=\frac{c}{3^{z-1}}=\frac 15$$
then
$$abc=\frac{3^{(x+y+z)}}{125}$$
then we need a condition for $t=x+y+z\in \mathbb R$, since $3^{t}$ can assume any value $\in(0,\infty)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3039091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find the sum of these variables. Five real numbers $a_1, a_2, a_3, a_4\;\text{and}\; a_5\;$ are such that
$$\sqrt{a_1- 1} + 2\sqrt{a_2- 4}+3\sqrt{a_3- 9}
+4\sqrt{a_4- 16} + 5 \sqrt{a_4- 25} =\frac{a_1+a_2+a_3+a_4+a_5}{2}.$$
Find $a_1+a_2+a_3+a_4+a_5.$
Thanks for checking this out!
| For real $\sqrt{a-b^2},$ we need $a-b^2\ge0$
and for $b>0,$ and as $\sqrt{a-b^2}\ge0$ by AM-GM inequality,
$$\dfrac{(\sqrt{a-b^2})^2+(b)^2}2\ge b\sqrt{a-b^2}$$
the equality will occur if $\sqrt{a-b^2}=b$
$$\implies\dfrac{(\sqrt{a_1-1})^2+1^2+\cdots+(\sqrt{a_5-5^2})^2+5^2}2=\dfrac{a_1+a_2+a_3+a_4+a_5}2 \ge \sqrt{a_1-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3039644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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The number of integral solution of $\alpha+\beta+\gamma+\delta$=18 such that.. Question
The number of integral solution of the equation $$\alpha+\beta+\gamma+\delta=18$$, with the conditions:
$1\leq\alpha\leq5$; ${-2}\leq\beta\leq4$; $0\leq\gamma\leq5$ and $3\leq\delta\leq9$ is $k$. Find $k$.
$$Attempt$$
I tried to d... | Here is a proof using the method of generating functions. By adjusting the constraints we may write the problem as finding the number of integral solutions to
$$
\alpha'+\beta'+\gamma'+\delta'=16
$$
with $0\le\alpha'\leq 4, 0\leq \beta'\leq 6, 0\leq \gamma'\leq 5, 0\leq\delta'\leq 6$. The number of solutions is then t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3045331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Integer Solutions of the Equation $u^3 = r^2-s^2$ The question says the following:
Find all primitive Pythagorean Triangles $x^2+y^2 = z^2$ such that $x$ is a perfect cube.
The general solution for each variable are the following:
$$x=r^2-s^2$$
$$y=2rs$$
$$z=r^2+s^2$$
such that $\gcd(r,s) = 1$and $r+s \equiv 1 \pmod ... | We can find one or more Pythagorean triples for any odd leg $\ge 3$using a function of $(m,A)$:
$$\text{We can let }n=\sqrt{m^2-A}\text{ where }\lceil\sqrt{A}\space\rceil\le m\le \frac{A+1}{2}$$
This is useful in finding $x^3+B^2=C^2$ because we can plug in any odd cube $(A)$ an make a small finite search of values bas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3046479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Prove that $\sum_{k=0}^{2n} \binom {2n+k}{k} \binom{2n}{k} \frac{(-1)^k}{2^k} \frac{1}{k+1} = 0. $
Let $n$ be a positive integer. Prove that
$$
\sum_{k=0}^{2n} \binom {2n+k}{k} \binom{2n}{k} \frac{(-1)^k}{2^k} \frac{1}{k+1} = 0.
$$
I am trying to solve this by using induction on $n$. I have proven the sum to be zer... | Starting from
$$\sum_{k=0}^{2n} {2n+k\choose k} {2n\choose k}
\frac{(-1)^k}{2^k} \frac{1}{k+1}$$
we get
$$\frac{1}{2n} \sum_{k=0}^{2n} {2n+k\choose k+1} {2n\choose k}
\frac{(-1)^k}{2^k}
= \frac{1}{2n} \sum_{k=0}^{2n} {2n+k\choose 2n-1} {2n\choose k}
\frac{(-1)^k}{2^k}
\\ = \frac{1}{2n} \sum_{k=0}^{2n} {2n\choose k}
\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3046755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
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How would I go about solving for $x$ in $\frac{(x-a)\sqrt{x-a}+(x-b)\sqrt{x-b}}{\sqrt{x-a}+\sqrt{x-b}}=a-b$? The question
This is a homework question. Given the following, I am to solve for $x$ in terms of $a$ and $b$:
$$\frac{(x-a)\sqrt{x-a}+(x-b)\sqrt{x-b}}{\sqrt{x-a}+\sqrt{x-b}}=a-b;a>b.$$
My attempt
Although I see ... | Hint: Define $$u=\sqrt{x-a}\\w=\sqrt{x-b}$$therefore $${w^3+u^3\over u+w}=w^2-u^2$$which yields to $$2u^3=uw^2-u^2w$$one answer is $u=0$ or $x=a$ which is valid. The others can be found by solving $$2u^2=w^2-uw$$or $$u^2+uw=a-b$$by substituting we obtain $$x-a+\sqrt{(x-a)(x-b)}=a-b$$
| {
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Find the limit of $\lim_{n\to\infty}((n^3+n^2)^{1/3}-(n^3+1)^{1/3})$ without using the identity $a^3-b^3=(a-b)(a^2+ab+b^2)$ Find the limit of the sequence $$\lim_{n\to\infty}((n^3+n^2)^{1/3}-(n^3+1)^{1/3})$$
I showed that the limit is $1/3$, using the identity $$a^3-b^3=(a-b)(a^2+ab+b^2)$$
we get the sequence is equal ... | 1) $n(1+1/n )^{1/3} = $
$\dfrac{(1+1/n)^{1/3}}{1/n}.$
2) $n(1+1/n^3 )^{1/3} =$
$\dfrac{(1+1/n^3)^{1/3}}{1/n}.$
$\small{\dfrac{((1+1/n)^{1/3} -1) -((1+1/n^3)^{1/3}-1)}{1/n}}$
$\small{=\dfrac{(1+1/n)^{1/3} -1}{1/n} - (1/n^2)\dfrac{(1+1/n^3)^{1/3} -1}{1/n^3}.}$
First term:
Let $f(x)=x^{1/3}$:
$\lim_{n \rightarrow \infty}\... | {
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Closed form of this type $\sum_{j=0}^{\infty}\frac{2^jj^n}{(2j+1)(2j+3){2j \choose j}}$ Given that, $$\sum_{j=0}^{\infty}\frac{2^j\left(j-\frac{1}{3}\right)^3\left(j^2+j-1\right)}{(2j+1)(2j+3){2j \choose j}}=A\tag1$$
We have $A=2\pi+12+\frac{1}{3}?$
We can generalize the above $(1)$:
$$\sum_{j=0}^{\infty}\frac{2^jj^n}{... | From the expansion of $\arcsin^2 t$
\begin{equation}
\arcsin^2 t=\sum_{p=0}^\infty \frac{2^{2p}t^{2p+2}}{(2p+1)(p+1)\binom{2p}{p}}
\end{equation}
we can obtain by differentiation
\begin{equation}
2\frac{\arcsin t}{\sqrt{1-t^2}}=\sum_{p=0}^\infty \frac{2^{2p+1}t^{2p+1}}{(2p+1)\binom{2p}{p}}
\end{equation}
Multiplyi... | {
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About the equivalence relation on $\mathbb{Z}\times\mathbb{Z}\,$ s.t. $(a,b)\sim(c,d)\,$ if $\,2^{a^2+d^2}\equiv 2^{b^2+c^2} (\text{mod} \, 5)$ Let $\sim$ be an equivalence relation on $\mathbb{Z}\times\mathbb{Z}\,\,\,\,\,$ s.t. $\,\,\,\,\,\forall \,\,\,(a,b), (c,d)\in\mathbb{Z}\times\mathbb{Z}$:
$(a,b)\sim(c,d)\,\,\,\... | Since $\gcd(2,5)=1$ and $2^4 \equiv 1 \pmod{5}$, we have
$$2^{a^2+d^2} \equiv 2^{b^2+c^2} \pmod{5} \implies a^2+d^2 \equiv b^2+c^2 \pmod{4}\implies a^2-b^2 \equiv c^2-d^2 \pmod{4}.$$
Actually all these implications are reversible. So
$$2^{a^2+d^2} \equiv 2^{b^2+c^2} \pmod{5} \iff a^2-b^2 \equiv c^2-d^2 \pmod{4}.$$
This... | {
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Rolling five $6$-sided dice, where (as an example) $3$, $2$ and $1$ faces of each die are equivalent Let's say we have a $6$ sided die. $3$ of the sides have the value $A$, $2$ have the value $B$ and $1$ has the value $C$.
If we take $5$ of those, roll them together and look at the possible values, how would I go about... | Someone else said use the multinomial distribution, but to be a little more clear:
If you have values $v_1,v_2,\ldots, v_k$ with multiplicity numbers $m_1,m_2,\ldots, m_k$ with $\sum_{i=1}^k m_i = M$, then the probability that a single fair die will roll value $v_i$ is $\dfrac{m_i}{M}$.
So, if you roll $n$ dice and you... | {
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greatest common divisor of two elements Find all possible values of GCD(4n + 4, 6n + 3) for naturals n
and prove that there are no others
3·(4n + 4) - 2·(6n + 3) = 6,
whence the desired GCD is a divisor 6.
But 6n + 3 is odd, so only 1 and 3 remain.
n=1 and n=2 are examples for GCD=1 and GCD=3
is the solution correct... | Your way is correct.
Other way.
$\gcd(4n+4, 6n+3) = \gcd(4n+4, (6n+3) - (4n+4)) =$
$\gcd (4n+4, 2n -1) = \gcd(4n+4 - 2(2n-1), 2n-1)=$
$\gcd (6, 2n- 1) = $
... Now two things should be apparent. $2n-1$ is odd and $6$ is even so the prime factor $2$ of $6$ will not be a factor of $2n-1$. And Lemma: if $\gcd(j,b) = 1$... | {
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How does this equation always work? $\sqrt{x \times (x+2)+1} = x + 1$ for non-negative $x$, and $=|x|-1$ for negative $x$ When I was playing with new calculator's functions, somehow I managed to get a formula, which works with all real numbers (negative number have a slight change). I had asked my teacher about it, but... | Note that you have a perfect square trinomial.
$$x(x+2)+1 = x^2+2x+1$$
You probably know that by binomial expansion, $(a\pm b)^2 = a^2\pm 2ab+b^2$.
$$(a\pm b)^2 = (a\pm b)(a\pm b) = a^2\pm ab\pm ba +b^2 = a^2\pm2ab+b^2$$
Notice the trinomial $x^2+2x+1$. You can see there are two perfect squares: $x^2$ and $1$. Also, th... | {
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a problem on complex numbers Let $w\neq 1$ and $w^{13} = 1$.
If $a = w+ w^3 + w^4 + w^{-4} + w^{-3} + w^{-1}$ and $b = w^2+ w^5 + w^6 + w^{-6} + w^{-5} + w^{-2}$, then the quadratic equation whose roots are $a$ and $b$ is ... ?
I got $w=\cos(\frac{2\pi}{13})+i\sin(\frac{2\pi}{13})$
And then I found $a$ and $b$ in tri... | Step 1: the equation you want is $(z-a)(z-b)=0$. Expand the product and you get $$z^2-(a+b)z+ab=0$$
Step 2: Use $w^{13}=1$, so $w^{-1}=w^{13}w^{-1}=w^{12}$ similarly, for all negative powers $$w^{-n}=w^{13-n}$$
Step 3: $$a+b=w+w^3+w^4+w^9+w^{10}+w^{12}+w^2+w^5+w^6+w^7+w^8+w^{11}=\\=\frac{w^{13}-1}{w-1}-1=-1$$
Step 4: T... | {
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Prove that $\frac{1}{2}(x+\frac{a}{x}) \ge \sqrt{a}$ Let x and a be real numbers > 0. Prove that $\frac{1}{2}(x+\frac{a}{x}) \ge \sqrt{a}$
My idea is that I'm going to use $a>b \iff a^2>b^2$ since we are only dealing with postive real numbers we won't run into problems with the root.
$\frac{1}{2}(x+\frac{a}{x}) \ge \sq... | From here:
$$\frac{1}{4}x^2 +\frac{1}{2}a+\frac{a^2}{4x^2} \ge a,$$
move the $a$ to the left side to yield
$$\frac{1}{4}x^2 - \frac{1}{2}a+\frac{a^2}{4x^2} \ge 0.$$
The left side is a perfect square:
$$\left(\frac{x}{2} - \frac{a}{2x}\right)^2.$$
| {
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Determinant of matrix of submatrices Can you check my solution to:
Task
Having two matrices $X,Y\in\mathbb{R}^{n,n}$ where $x,y\in\mathbb{R}$ and matrices are defined as
$
X=\begin{bmatrix}
x & 0 &0 & \dots & 0 \\
x & x & 0& \dots & 0\\
x & x & x& \dots & 0\\
\vdots&\vdots&\vdots&\vdots&\v... | Your derivation of the main result (that the determinant is $(x^2+y^2)^n$) is correct, but your verification is not. You should have $\det\pmatrix{0&Y\\ -Y&0}=\det(Y)^2=y^{2n}$. E.g. if you interchange the first and the last column of $\pmatrix{0&Y\\ -Y&0}$, you get a factor of $-1$. However, the entry $-y$ also contai... | {
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Trying to simplify $\frac{\sqrt{8}-\sqrt{16}}{4-\sqrt{2}} - 2^{1/2}$ into $\frac{-5\sqrt{2}-6}{7}$ I'm asked to simplify $\frac{\sqrt{8}-\sqrt{16}}{4-\sqrt{2}} - 2^{1/2}$ and am provided with the solution $\frac{-5\sqrt{2}-6}{7}$
I have tried several approaches and failed. Here's one path I took:
(Will try to simplify ... | You were doing fine until the place where you tried to expand
$(2\sqrt2 - 4)(4 + \sqrt2).$
There are mnemonic techniques for this but I think plain old distributive law works well enough:
\begin{align}
(2\sqrt2 - 4)(4 + \sqrt2) &= (2\sqrt2 - 4)4 + (2\sqrt2 - 4)\sqrt2 \\
&= (8\sqrt2 - 16) + (4 - 4\sqrt2) \\
&= 4\sqr... | {
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Polynomial Long Division Confusion (simplifying $\frac{x^{5}}{x^{2}+1}$) I need to simplify \begin{equation}
\frac{x^{5}}{x^{2}+1}
\end{equation}
by long division in order to solve an integral.
However, I keep getting an infinite series:
\begin{equation}
x^{3}+x+\frac{1}{x}-\frac{1}{x^{3}}+...
\end{equation}
| The idea of polynomial division is like integer division. With integer division of $\frac nd$, we want integer $q,r$ so that $n=qd+r$ and $r\lt d$. With polynomial division of $\frac nd$, we want polynomial $q,r$ so that $n=qd+r$ and $\deg(r)\lt \deg(d)$.
$$
\require{enclose}
\begin{array}{rl}
&\phantom{)\,}\color{#C00... | {
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Minimum value of $\frac{4}{4-x^2}+\frac{9}{9-y^2}$ Given $x,y \in (-2,2)$ and $xy=-1$
Minimum value of $$f(x,y)=\frac{4}{4-x^2}+\frac{9}{9-y^2}$$
My try:
Converting the function into single variable we get:
$$g(x)=\frac{4}{4-x^2}+\frac{9x^2}{9x^2-1}$$
$$g(x)=\frac{4}{4-x^2}+1+\frac{1}{9x^2-1}$$
Using Differentiation we... | From $xy=-1$
$$ S = \frac{4}{4-x^2} + \frac{9}{9-y^2} = 1 + \frac{35}{25 - \left(3x - \frac{2}{x}\right)^2}\geq \frac{12}{5}$$
Equality hold when $$3x=\frac{2}{x}$$
| {
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Equation of auxiliary circle of the ellipse $2x^2 +6xy + 5y^2$ =1 Equation of auxiliary circle of the ellipse $2x^2 +6xy + 5y^2$= 1
My approach is , First I try remove xy term from the equation, to convert the given equation in the standard equation of ellipse and find the value of $a$ and $b$. For this , I use the con... | The auxiliary circle is centered at the center of the ellipse and its radius equals the semi-major axis of the ellipse. Any conic of the form $ax^2+2hxy+by^2=1$ is centered at origin (as if $(x,y)$ lies on the ellipse so does $(-x,-y)$).
To find the semi-major axis we need to find the greatest distance of any point on ... | {
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Nigerian Olympiad Suppose $a,b,c,d$ are integers satisfying $ab + cd = 44,ad - bc = 9.$
Find the minimum possible value of $a² + b² + c² + d².$
| $$(ab+cd)^2=44^2\implies a^2b^2+2abcd+c^2d^2=44^2$$
Similarly:
$$a^2d^2-2abcd+c^2b^2=9^2$$
So:
$$a^2b^2+a^2d^2+c^2d^2+c^2b^2=44^2+9^2=2017\implies(a^2+c^2)(b^2+d^2)=2017$$
$2017$ is prime so one of $a^2+c^2$ or $b^2+d^2$ is $1$ and the other is $2017$. Assume $a^2+c^2=1$. Then one of $a$ and $c$ is $0$ and the other ... | {
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Solving a Cauchy problem, differential equation I have the following Cauchy problem
\begin{cases} y'(x) + \frac{1}{x^2-1}y(x) = \sqrt{x+1} \\ y(0) = 0 \end{cases}
I proceed by finding $e^{A(x)} $ where $A(x)$ is the primitive of $a(x)= \frac{1}{x^2-1}$ :
$$\int A(x)dx=\int \frac{1}{x^2-1}dx= \frac{1}{2} \log\Big(\fra... | Computing $$\mu(x)=e^{\int\frac{1}{x^2-1}dx}=\frac{\sqrt{1-x}}{\sqrt{1+x}}$$ then you will get
$$\int\frac{d}{dx}\left(\frac{\sqrt{1-x}y(x)}{\sqrt{x+1}}\right)=\int\sqrt{1-x}dx$$
| {
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If $\cos^4 \alpha+4\sin^4 \beta-4\sqrt{2}\cos \alpha \sin \beta +2=0$, then find $\alpha$, $\beta$ in $(0,\frac\pi2)$
If $\cos^4 \alpha+4\sin^4 \beta-4\sqrt{2}\cos \alpha \sin \beta +2=0$,
where $\displaystyle \alpha, \beta \in \bigg(0,\frac{\pi}{2}\bigg)$. Then value of $\alpha,\beta$ are
Try: I am trying to conver... | If you just call $x=\cos(\alpha)$ and $y=\sin(\beta)$ you get the 4th degree polynomial equation $x^4-4\sqrt{2}xy + y^4 +2 =0$. I don't see a clever way to solve this directly but this can be solved algebraically. So I would put this equation into mathematica or some similar software and then look a the 4 solutions.
| {
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Induction proof for stirling of first kind. I have to show that for every stirling number of the first kind $\forall n \geq 2 : s_{n,n-2} = \frac{1}{24}n(n-1)(n-2)(3n-1) $ is true.
I've started like this:
Base case: Let $n$ be $n=2$, then per defintion
$s_{n,0} = 0$.
Since we have $s_{2,2-2} = s_{2,0} = 0$ and $\frac{1... | $s_{n+1,n-1} = \frac{1}{24}n(n-1)(n-2)(3n-1)+n*s_{n,n-1} \tag1$.
$s_{n,n-1} = \frac{n(n-1)}{2} \tag 2$.
Substitute this in the prior equation we get
$s_{n+1,n-1} = \frac{1}{24}n(n-1)(n-2)(3n-1)+\frac{n^2(n-1)}{2} $.
which simplifies to
$s_{n+1,n-1} = \frac{1}{24}(n+1)n(n-1)(3n+2)\tag 3 $.
Thus proved by induction. Yo... | {
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How to see $x^6-1=(x^2−1)(x^2+x+1)(x^2−x+1)$? I was reading an example where the purpose was to compute a certain Galois group. Along the way, the writer says : note $x^6-1=(x^2−1)(x^2+x+1)(x^2−x+1)$. But how do I note this? I understand you can factorize by $x^2-1$, since when I draw on the unit circle I see that $-1$... | $x^6-1=(x^3+1)(x^3-1)\\x^3-1=(x-1)(x^2+x+1)\\x^3+1=(x+1)(x^2-x+1)$ might help.
| {
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Find all positive triples of positive integers a, b, c so that $\frac {a+1}{b}$ , $\frac {b+1}{c}$, $\frac {c+1}{a}$ are also integers. Find all positive triples of positive integers a, b, c so that $\frac {a+1}{b}$ , $\frac {b+1}{c}$, $\frac {c+1}{a}$ are also integers.
WLOG, let a$\leqq b\leqq c$,
| If any two of $a,b,c$ are equal, then wlog. $a=b$. As $\frac{b+1}{a}=1+\frac1a$ is an integer, we conclude $a=b=1$. The remaining conditions are that $\frac{c+1}{1}$ and $\frac 2c$ are integers, which lead us to the solutions
$$(1,1,1),\qquad (1,1,2) $$
(and cyclic permutations of the latter).
So assume $a,b,c $ are pa... | {
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Expectation and variance of the number of elements of a random non-empty set selected from a finite power set
Let $S$ denote a finite set of cardinality $|S| = N$. Select randomly a non-empty subset of $S$. Let $X$ indicate the number of items belonging to this subset.
(a) Describe the probability mass function of $X$... | Seem probably simpler if you look at it this way: Say $S$ has $n$ elements. If $Y$ is the number of elements of a randomly selected element of the power set, including the empty set as a possibility, then $Y$ has the same distribution as $X_1+\dots+X_n$, where the $X_j$ are iid with $P(X_1=0)=P(X_1=1)=1/2$.
| {
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Probability of $2$ pairs on $5$ dice We throw $5$ dice. What is the probability of getting $2$ pairs ?
My solution says it is $$\frac{6\cdot 5\cdot 4\cdot 5! }{6^5\cdot 2\cdot 2!\cdot 2!},$$ where as for me it's $$\frac{6\cdot 5\cdot 4\cdot 5!}{6^5\cdot 2!\cdot 2!}.$$
I do as follows: Throwing $5$ dice is the same thin... | You have to divide your result by $2$ because you are not interested in the order of the two different pairs: $11336$ is the same of $33116$.
We have $\binom{6}{2}$ ways to choose the values of the two couples (say $A$ and $B$ with $A<B$) and $4$ ways to choose the single value (say $C$). The number of permutations of... | {
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Creating an integral to represent the volume of the intersection of two balls in cartesian coordinates The question states:
Let $A$ be the intersection of the balls
$x^2+y^2+z^2\leq 9$ and $x^2+y^2+(z-8)^2\leq 49$
I am asked to just set up the iterated triple integral that represents the volume of $A$ in cartesian coor... | $\def\bbr{\mathbf{R}}$It is simpler to exploit the symmetries. From the picture below, you can see clearly that the intersection is a normal domain in $\bbr^3$, which means that you can write it as
$$
A=\{(x,y,z)\mid (x,y)\in D, f(x,y)\le z\le g(x,y)\}
$$
where $D$ is some two-dimensional region on the $xy$-plane. You ... | {
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What's the answer to $\int \frac{\cos^2x \sin x}{\sin x - \cos x} dx$? I tried solving the integral $$\int \frac{\cos^2x \sin x}{\sin x - \cos x}\, dx$$ the following ways:
*
*Expressing each function in the form of $\tan \left(\frac{x}{2}\right)$, $\cos \left(\frac{x}{2}\right)\,$ and $\,\sin \left(\frac{x}{2}\righ... | Hint:
Let $\dfrac\pi4-x=y$
$\sin x-\cos x=\sqrt2\sin y$
$\sin x=\dfrac{\cos y-\sin y}{\sqrt2}$
$2\cos^2x=1+\cos2x=1+2\sin y\cos y$
$$\dfrac{(\cos y-\sin y)(1+2\sin y\cos y)}{\sin y}=2\cos^2y-2\sin y\cos y+\cot y-1=\cos2y-\sin2y+\cot y$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3082128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 2
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A Problem related to the Cauchy Distribution The following problem is from the book, "Introduction to Probability" by
Hoel, Port and Stone. My answer does not match the back of the book. What did I do wrong?
Thanks,
Bob
Problem:
Let $X$ have a Cauchy density. Find the upper quartile for $X$.
Answer:
Recall the density ... | You forgot the factor $\frac{1}{\pi}$ in the probability density function. Correct density function is
$$
f(x) =\frac{1}{\pi(1+x^2)}
$$ so that $\int_{-\infty}^\infty f(x) dx =1$. Your problem is to find $a$ such that
$$
\frac{1}{\pi}\int_{-\infty}^a \frac{dx}{1+x^2}=\frac{\arctan a+\frac{\pi}{2}}{\pi}=\frac{3}{4}
$$ o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3082241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Geometry with circle and triangle. Any other solutions(advice) are welcome.
What I am asking is this because I am studying mathematics through feedback process of my solution and learning new solutions.
Please, release Hold on.
When will the hold be resolved? I want to see your new solutions(or advices).
$\overline{AE}... |
Here is a completely different soluiton based on trigonometry.
From triangle $ABD$:
$$\frac{3x}{\sin\alpha}=\frac{l}{\sin(90^\circ-\beta)}\tag{1}$$
From triangle $ADC$:
$$\frac{2x}{\sin\beta}=\frac{l}{\sin(90^\circ-\alpha)}\tag{2}$$
From triangle $ABC$:
$$\frac{1}{\sin(90^\circ-\alpha)}=\frac{2}{\sin(90^\circ-\beta)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3084169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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$\left\lfloor \frac{a-b}{2} \right\rfloor + \left\lceil \frac{a+b}{2} \right\rceil = a$ when $a,b$ are integers? Let $a$ and $b$ be positive integers.
If $b$ is even, then we have $$\left\lfloor \frac{a-b}{2} \right\rfloor + \left\lceil \frac{a+b}{2} \right\rceil = a$$
I think the equality also hold when $b$ is odd. Wh... | If $a$ and $b$ are both odd then we have $a=2m+1$ and $b=2n+1$ where $m$ and $n$ are positive integers.
Then, we have \begin{align}\left\lfloor\frac{a-b}2\right\rfloor+\left\lceil\frac{a+b}2\right\rceil &= \left\lfloor\frac{(2m+1)-(2n+1)}2\right\rfloor+ \left\lceil\frac{(2m+1)+(2n+1)}2\right\rceil\\
&=\left\lfloor\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3087071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
If $\cos^6 (x) + \sin^4 (x)=1$, find $x$ if $x\in [0, \dfrac {\pi}{2}]$ If $\cos^6 (x) + \sin^4 (x)=1$, find $x$ if $x\in [0, \dfrac {\pi}{2}]$
My attempt:
$$\cos^6 (x) + \sin^4 (x)=1$$
$$\cos^6 (x) + (1-\cos^2 (x))^{2}=1$$
$$\cos^6 (x) + 1 - 2\cos^2 (x) + \cos^4 (x) = 1$$
$$\cos^6 (x) + \cos^4 (x) - 2\cos^2 (x)=0$$
| Hint:
Clearly, $\sin x=0\iff\cos^2x=?,$
$\cos x=0\iff\sin^2x=?$ are solutions
For $0< a<1,$ $$a^6<a^4<a^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3087249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Calculate $\lim\limits_{n\rightarrow \infty }\sum_{k=1}^{n}\frac{6}{k(k+1)(k+3)}$ $$\lim_{n\rightarrow \infty }\sum_{k=1}^{n}\frac{6}{k(k+1)(k+3)}$$
I tried to simplify the sum and I got $\frac{2}{k}-\frac{3}{k+1}+\frac{1}{k+3}$ but I can't use this to simplify the terms.Also,I tried to amplify with $k+2$ and I got $$... | Hint
$$\frac{2}{k}-\frac{3}{k+1}+\frac{1}{k+3}=\frac{2}{k}-\frac{2}{k+1}-\frac{1}{k+1}+\frac{1}{k+3}=2\left(\frac{1}{k}-\frac{1}{k+1} \right)-\left(\frac{1}{k+1}-\frac{1}{k+3} \right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3089290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Find all integer solutions to $ x^3 - y^3 = 3(x^2 - y^2) $ The objective is to find all solutions to
$$ x^3 - y^3 = 3(x^2 - y^2) $$
where $x,y \in \mathbb{Z}$.
So far I've got one pair of solution. Try $(x, y)=(0,0)$:
$$ 0^3-0^3=3(0^2-0^2) \\
0=0 \qquad \text{equation satisfied}$$
Another try $ (x, y) = (x, x)$ then ... | $x=y$ and $x=3$ with $y=0$ (or $x=0$ with $y=3$) are trivial answers. To show there are no others for $x\ne y$, let $n=x+y$, then the expression can be written $x^2+xy+y^2=(x+y)^2 +(3-n)n$ or $x(n-x)+3n-n^2=0$. Solve the quadratic for $x$ to get $x=\frac{n+\sqrt{12n-3n^2}}{2}$. Since the discriminant has to be real, $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Show vector is approximately an eigenvector of matrix, thus find eigenvalue Say we have matrix $\mathbf{A}$
$$
\mathbf{A}=\begin{pmatrix}
-3&2&0\\
4&-6&2\\
0&1&-1
\end{pmatrix}
$$
We now must show that $\mathbf{v}=\begin{pmatrix}-1.34&-0.8&1\end{pmatrix}^T$ is an approximate eigenvector for $\mathbf{A}$ to 2 decimal pl... | Your method has a minor issue.
If you want numbers to match to a certain precision, you should use inequalities, e.g., solve
$$|-1.34\lambda-2.42|<0.01,$$
$$|-0.8\lambda-1.44|<0.01,$$
$$|1\cdot\lambda-(-1.8)|<0.01.$$
If you want it to match exactly to 2 decimal places, you may need to shrink these margins (they are n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3090830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Evaluate $\int_{0}^{1}\frac{1+x+x^2}{1+x+x^2+x^3+x^4}dx$ Evaluate $$I=\int_{0}^{1}\frac{(1+x+x^2)}{1+x+x^2+x^3+x^4}dx$$
My try:
We have:
$$1+x+x^2=\frac{1-x^3}{1-x}$$
$$1+x+x^2+x^3+x^4=\frac{1-x^5}{1-x}$$
So we get:
$$I=\int_{0}^{1}\frac{1-x^3}{1-x^5}dx$$
$$I=1+\int_{0}^{1}\frac{x^3(x^2-1)}{x^5-1}dx$$
Any idea from her... | This integral appeared on AoPS a while ago, it might be that it was on MSE too. Anyway I will quote what I did on AoPS.
$$I=\int_{0}^{1}\frac{(1+x+x^2)}{1+x+x^2+x^3+x^4}dx=\int_0^1 \frac{1-x^3 }{1-x^5}dx$$ Recall the geometric series: $\displaystyle{\frac{1}{1-x^z}=\sum_{n=0}^\infty x^{nz}, \, |x|<1 } $. Applying it h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3090916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 2
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Evaluate ${\lim_{x \rightarrow 0} \frac{1-\cos x\cos 2x\cdots \cos nx}{x^2}}$
Evaluate $${\lim_{x \rightarrow 0} \frac{1-\cos x\cos 2x\cdots \cos nx}{x^2}}$$
It should be $$\frac{1}{12}n(n+1)(2n+1)$$ but I don't know how to prove that. I am also aware that $\lim\limits_{\theta \to 0} \dfrac{1-\cos \theta}{\theta ^2}=... | This is handled in the same manner as in this answer.
Split the numerator like $$1-\cos x+\cos x(1-\cos 2x\cos 3x\dots\cos nx) $$ and the desired limit is equal to $$\lim_{x\to 0}\frac {1-\cos x} {x^2}+\lim_{x\to 0}\cos x\cdot \frac{1-\cos 2x\dots\cos nx} {x^2}$$ which is same as $$\frac {1} {2}+\lim_{x\to 0} \frac{1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3092676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
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Calculating limit of definite integral I need to calculate:
$$
\lim_{x\to \infty} \int_x^{x+1} \frac{t^2+1}{t^2+20t+8}\, dt
$$ The result should be $1$.
Is there a quicker way than calculating the primitive function?
I thought about seperating to $\int_0^{x+1} -\int_0^x$ but still can't think of the solution.
| Use the estimate:
$$\frac{t-10}{t+10}<\frac{t^2+1}{t^2+20t+8}<\frac t{t+10}, \ t>1 \Rightarrow \\
\int_x^{x+1} \frac{t-10}{t+10}dt<I(x)<\int_x^{x+1} \frac t{t+10}dt \Rightarrow \\
1-20\ln \frac{x+11}{x+10}<I(x)<1-10\ln \frac{x+11}{x+10} \Rightarrow \\
\lim_{x\to\infty} \left(1-20\ln \frac{x+11}{x+10}\right)\le \lim_{x\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3094982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 5
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How to solve $2x(4-x)^{\frac{-1}{2}}-3\sqrt{4-x}=0$ for $x$? I'm struggling to figure out how to solve $2x(4-x)^{\frac{-1}{2}}-3\sqrt{4-x}=0$ for $x$.
The answer is $x = \frac{12}{5}$, but I am getting $x = \frac{-35}{9}$
My steps are:
$2x(4-x)^{\frac{-1}{2}}-3\sqrt{4-x}=0$
*
*make exponent postive
$2x\frac{1}{\s... | If $$\frac{2x}{\sqrt{4-x}}-3\sqrt{4-x}=0$$
Then
$$\frac{2x}{\sqrt{4-x}}=\frac{3\sqrt{4-x}}{1}$$
If we cross multiply we get that
$$2x=3(4-x)=12-3x$$
You can go from here....
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Find expected value of a random rectangle in a square grid I've tried to solve the following problem. Could you please tell if I did it right.
Problem. Given a n x n square grid. We take a random rectangle so that every ractangle is equally likely.
Find expected value of area of this rectangle.
Solution.
The number of ... | The expected area is the expected base times the expected height. If, for example, $n=10$, there are $10$ bases of length $1$, and $9$ of length $2$, on up to $1$ of length $n$: $\sum_{k=1}^n k$ in all. Thus the expected base length is $$b = \frac{\sum_{k=1}^n k ( n+1-k)}{\sum_{k=1}^n k}$$ and the expected height leng... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3096871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Integrate using trigonometric substitution. Am I on the right path? I have been trying to solve:
$$\int \frac{\sqrt{x^2-9}}{x^3} dx$$
I am letting $ x = 3\sec \theta$ and so $dx = 3 \sec \theta \tan \theta$
So then I have:
$$\int \frac{\sqrt{9\sec^2 \theta - 9}}{27 \sec^3 \theta} dx$$
$$\int \frac{\sqrt{9(\sec^2 \thet... | $$I=\int\frac{\sqrt{x^2-9}}{x^3}dx=\int\frac{3\sqrt{(x/3)^2-1}}{x^3}dx$$
and we know that $\cosh^2\theta-1=\sinh^2\theta$
$$x=3\cosh(y)$$
$$dx=3\sinh(y)dy$$
$$I=9\int\frac{\sqrt{\cosh^2(y)-1}}{27\cosh^3(y)}\sinh(y)dy=\frac13\int\frac{\sinh^2(y)}{\cosh^3(y)}dy=\frac13\int\text{sech}(y)-\text{sech}^3(y)dy$$
a reduction f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3098752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Factorise $x^n + 1$ Is there a way to factorise $x^n + 1$
I thought of doing it like this: $$x^n +1 = (x+1)(x^{n-1} - x^{n-2} + \cdots - x + 1)$$ but can't seem to get anywhere using this method.
| If $n$ is even then the graph of $y=x^n+1$ is always above the $x$ axis so $x^n+1=0$ has no real roots.
If $n$ is odd then the graph of $y=x^n+1$ crosses the $x$ axis at $x=-1$ and cannot cross the $x$ axis anywhere else because $x^n+1$ is monotonically increasing. So $x^n+1=0$ has a single real root at $x=-1$, and so ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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What is the value of $m+n$ if $\frac{m}{n}$ is the radius of the smallest of the three circles?
Circles of radii 5, 5, 8 and $\frac{m}{n} $(the smallest circle) are
mutually externally tangent to all circles, where $m$ and $n$ are
relatively prime positive integers. Find $m + n$.
Source: Bangladesh Math Olympia... | Let $x$ be a radius of the little circle.
Thus, by Heron we can get an area of the left and of the right triangle:
$$\sqrt{(x+13)x\cdot8\cdot5}=\sqrt{40x(x+13)}.$$
An area of the lower triangle it's
$$\frac{10\sqrt{(x+5)^2-5^2}}{2}=5\sqrt{x^2+10x)}.$$
The area of the full triangle it's
$$\frac{10\sqrt{13^2-5^2}}{2}=60.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3103900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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A sum of series problem with alternating sign of terms I came across a problem that requires me to find the sum of a series. The term of the series $T_n$ is given by
$$T_n = (-1)^{\frac{n(n+1)}2}n^2$$
Sum till $4n$ terms is to be found.
Writing down the first few terms:
$$-(1)^2-(2)^2+3^2+4^2-(5)^2-(6)^2+7^2+8^2+ \ldot... | If the number of terms is multiple of$4,$ $$\sum_{r=0}^n(-(4r+1)^2-(4r+2)^2+(4r+3)^2+(4r+4)^2)=2\sum(8r+4+8r+6)=?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3110394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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How to compute remainder of division of $P(x)$ by $x^2 -3x+2$?
The remainder of division of $P(x)$ by $x^2−1$ is $2x+1$, and the remainder of division of the same polynomial by $x^2−4$ is $x+4$. Compute the remainder of division of $P(x)$ by $x^2−3x+2$.
I will translate these into math equations
$$P(x) = (x^2-1)Q(x)+... | HINT:
$$\frac{P(x)}{x^2-1}=Q(x)+\frac{2x+1}{x^2-1}\implies\frac{P(x)}{x-1}=[(x+1)Q(x)+2]+\frac 3{x-1}\\\frac{P(x)}{x^2-4}=R(x)+\frac{x+4}{x^2-4}\implies\frac{P(x)}{x-2}=[(x+2)R(x)+1]+\frac6{x-2}\\\boxed{\frac{P(x)}{x-2}-\frac{P(x)}{x-1}=\frac{(x-1)P(x)-(x-2)P(x)}{x^2-3x+2}=\frac{P(x)}{x^2-3x+2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3110509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Triple integral : volume I need to calculate the volume between $x^2+y^2\le z^2-1$ and $2x^2+y^2+z^2\le 2$.
So It's a hyperboloid of two-sheets intersected with an ellipsoid.
their intersection leads to: $3x^2+2y^2=1$ which is an ellipse.
Using Cylindrical coordinates : $(x,y,z)=(\frac{1}{\sqrt{3}}r\cos\theta,\frac{1}{... | The thought process was correct.
Hint: the integrand should be $\frac{\sqrt{6}}{6}r^2$.
Not $\frac{\sqrt{6}}{6}r$, since the Jacobian is $\frac{\sqrt{6}}{6}r$ for the change of variables.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3111332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Stuck at proving whether the sequence is convergent or not I have been trying to determine whether the following sequence is convergent or not. This is what I got:
Exercise 1: Find the $\min,\max,\sup,\inf, \liminf,\limsup$ and determine whether the sequence is convergent or not:
$X_n=\sin\frac{n\pi}{3}-4\cos\frac{n\... | If the sequence is convergent, then all of its subsequences should
converge to the same limit.
Consider the subsequence defined by $n_{k}=3k$:
$$
X_{n_{k}}=\sin(k\pi)-4\cos(k\pi)=-4\cos(k\pi)=\begin{cases}
-4 & \text{if }k\text{ is even}\\
+4 & \text{if }k\text{ is odd}.
\end{cases}
$$
This subsequence oscillates betwe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3114035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 3
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Is there any natural number triangle whose inscribed circle's radius is $1$ except length $(a,b,c)=(3,4,5)$?
Is there any natural number triangle that inscribed circle's radius is $1$ except length $(a,b,c)=(3,4,5)$?
I found that there are no right triangle except $(3,4,5)$.
Thm. There are only one natural number r... | Say $a\leq b\leq c$. If $s=(a+b+c)/2$ then $$r={S\over s}$$ where $S$ is area. By Heron formula we have
$$S=s \implies \boxed{(s-a)(s-b)(s-c)=s}$$
then $$s-a\mid s\implies s-a\mid a \implies a = k(s-a)$$
so $$k(b+c)=a(2+k)$$ Since $b+c\geq 2a$ we get $$2ka\leq a(2+k)\implies k\leq 2$$
*
*$k= 1$ we have $b+c=3a$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3116044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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conversion of Binomial identity into series sum Prove that $$\binom{n}{1}(1-x)-\frac{1}{2}\binom{n}{2}(1-x)^2+\frac{1}{3}\binom{n}{3}(1-x)^3+\cdots \cdots +(-1)^{n-1}\frac{1}{n}(1-x)^n$$
$$=(1-x)+\frac{1}{2}(1-x^2)+\frac{1}{3}(1-x^3)+\cdots +\frac{1}{n}(1-x^n)$$
what i try
$$\bigg[1-(1-x)\bigg]^n=\binom{n}{0}-\binom{n}... | We put
$$S_n(x) = \sum_{p=1}^n \frac{1}{p} {n\choose p} (-1)^{p+1}
(1-x)^p.$$
Working first with the coefficient on $[x^q]$ where $1\le q\le n$
we see that it is
$$\sum_{p=q}^n (-1)^{p+1} \frac{1}{p}
{n\choose p} {p\choose q} (-1)^q.$$
Now
$${n\choose p} {p\choose q} =
\frac{n!}{(n-p)! \times q! \times (p-q)!}
= {n\ch... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3117115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find $a,b,c\in\mathbb{C}$, which minimize the value of the integral $\int_{-1}^1 |x^3-a-bx-cx^2|^2dx.$ We're working in the Hilbert Space $L^2([-1,1])$, we have $f(x)=x^3$ and $g(x)=cx^2 + bx + a$ with inner product
$$\langle f , g \rangle = \int_{-1}^1 |x^3-a-bx-cx^2|^2dx.$$
How does one go about finding the minimum?... | This is a simple least squares problem. You are looking for the projection of $f(x)=x^3$ on the subspace $S = span\{1,x,x^2\}$. The formula you presented is for $\|f-g\|^2$, not for $\langle f,g\rangle$. The coefficients $a,b,c$ will be the solutions of the linear system
$$
\begin{pmatrix}
\langle 1,1\rangle & \langle ... | {
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"url": "https://math.stackexchange.com/questions/3120182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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kangaroo maths competition How many three-digit positive integers $ABC$ exist, such that $(A + B)^c$ is a three-digit integer and an integer power of $2$?
Note: An integer power of $2$ is a number in the form $2^k$
, where $k$ is an integer.
(A) $15$
(B) $16$
(C) $18$
(D) $20$
(E) $21$
| Here is complete list of $21$ solutions.
*
*$2^7=128:$
*
*$2^7=(1+1)^7\implies ABC=117$
*$2^7=(2+0)^7\implies ABC=207$
*$2^8=4^4=16^2=256:$
*
*$2^8=(1+1)^8\implies ABC=118$
*$2^8=(2+0)^8\implies ABC=208$
*$4^4=(1+3)^4\implies ABC=134$
*$4^4=(2+2)^4\implies ABC=224$
*$4^4=(3+1)^4\implies ABC=314$
*$4^4=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3124646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Every real matrix $A$ is the linear combination of $4$ orthogonal matrices Question:
Prove that every matrix $A\in M_n(\mathbb R)$ is the linear combination of $4$ orthogonal matrices $X, Y, Z, W$ , i.e. $A=aX+bY+cZ+dW$ for some $a,b,c,d\in\mathbb R$.
This problem is taken from a forum and this is my paraphrase. It ... | In view of SVD, we may assume that $A\neq 0$ and $A = \operatorname{diag}(\lambda_1, \cdots, \lambda_n)$ such that $\lambda_i$'s are non-negative, and $\lambda_n$ is the largest among $\lambda_i$'s.
Under this assumption, we have $\lambda_n > 0$. Now write $\lambda_{2i-1} = a_i - b_i$ and $\lambda_{2i} = a_i + b_i$. I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3128869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
A mistake on computing $\int \frac{dx}{\sqrt{x+1}+1}$ I have to find $\int \frac{dx}{\sqrt{x+1}+1}$. This was my attempt, which is wrong and I cannot find where exactly is the mistake.
First I write $\frac{1}{\sqrt{x+1}+1}=\frac{\sqrt{x+1}-1}{x}=\frac{\sqrt{x+1}}{x}-\frac{1}{x}$, therefore $\int \frac{dx}{\sqrt{x+1}+1}... | But you have been right all along!
Notice that
$$\log\frac{\sqrt{x+1}-1}{\sqrt{x+1}+1} - \log x = \log\frac{\sqrt{x+1}-1}{x(\sqrt{x+1}+1)} = \log \frac{1}{(\sqrt{x+1}+1)^2} = -2\log(\sqrt{x+1}+1)$$
by your very first step.
Also, I would suggest that you use $\int\frac{dx}{x} = \log|x|$ (with the absolute value sign, to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3132934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
Derivative of a product of trig functions: $3\sin(x)\cot(x)$ I am having a lot of trouble understanding on how to find the derivative of $3\sin x\cot x$.
I end up with $-3\cos x/\sin^2 x$
| You need to use the product rule:
$$
(3\sin{x}\cot{x})'=\\
3(\sin{x})'\cot{x}+3\sin{x}(\cot{x})'=\\
3\cos{x}\cot{x}+3\sin{x}(-\csc^2{x})=\\
3\cos{x}\cot{x}-3\sin{x}\csc^2{x}=\\
3\cos{x}\frac{\cos{x}}{\sin{x}}-3\sin{x}\frac{1}{\sin^2{x}}=\\
3\frac{\cos^2{x}}{\sin{x}}-3\frac{1}{\sin{x}}=\\
3\frac{1}{{\sin{x}}}(\cos^2{x}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3133189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Does this pattern of summing polygonal numbers to get a square repeat indefinitely? I am using the table of polygonal numbers on this site:
http://oeis.org/wiki/Polygonal_numbers
The first row of the table gives the value of $n=0,1,2,3,...$. The first column gives the polygonal numbers $P_{N}(n)$ starting with $N=3... | I'm adding another answer because I wanted to try and use the formula for polygonal numbers as given here in an attempt to better answer the original question by the OP. In both cases above we exploited the arithmetical sequence properties and only used Triangular numbers. Can we get the same result strictly using th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3133859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
If nine coins are tossed, what is the probability that the number of heads is even? If nine coins are tossed, what is the probability that the number of heads is even?
So there can either be 0 heads, 2 heads, 4 heads, 6 heads, or 8 heads.
We have $n = 9$ trials, find the probability of each $k$ for $k = 0, 2, 4, 6, 8$
... | The easiest way to see this : Consider the number of heads we have in the first $8$ coins.
*
*If this number is even, we need a tail, we have probability $\frac{1}{2}$
*If this number is odd, we need a head, we have probability $\frac{1}{2}$
Hence no matter what the $8$ coins delivered, we have probability $\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3134991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "54",
"answer_count": 13,
"answer_id": 5
} |
Getting different answers when using product rule and limit substitution than I do with quotient rule I'm trying to differentiate $$y = \frac{x+1}{x-1}.$$
Using quotient rule:
http://prntscr.com/mt90yc
Using product:
http://prntscr.com/mt91dd
Using limit definition:
http://prntscr.com/mt91ih
I get $-2$ for product and ... | It means that you're doing something wrong.
$$
\left(\frac{x+1}{x-1}\right)'=
\frac{(x+1)'(x-1)-(x+1)(x-1)'}{(x-1)^2}=\\
\frac{x-1-(x+1)}{(x-1)^2}=
\frac{x-1-x-1}{(x-1)^2}=-\frac{2}{(x-1)^2}
$$
$$
[(x+1)(x-1)^{-1}]'=\\
(x+1)'(x-1)^{-1}+(x+1)[(x-1)^{-1}]'=\\
(x-1)^{-1}+(x+1)(-1)(x-1)^{-2}=\\
\frac{1}{x-1}-\frac{x+1}{(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3135426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Find value of $\frac{\sum AA_1 \cos\left(\frac{A}{2}\right)}{\sum \sin A}$ Triangle $\Delta ABC$ is inscribed in a circle of radius one unit. If the internal angle bisectors of angles $\angle A, \angle B,\angle C$ meets the circle at the points $A_1,B_1,C_1$ respectively. Find value of $$S=\frac{\sum AA_1 \cos\left(\fr... | Note that $A_1,$ $B_1,$ $C_1$ are the midpoints of $\overset{\huge\frown}{BC},$ $\overset{\huge\frown}{CA},$ $\overset{\huge\frown}{AB}$ respectively.
Therefore, considering $\triangle A_1BC,$ $A_1B=A_1C$. Now we can use law of cosines in $\triangle AA_1B$ and $\triangle AA_1C$ to get,
$$(A_1B)^2=(AA_1)^2+c^2-2\cdot AA... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3135557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Solving $\int_0^\frac{\pi}{2} \frac{\sqrt[3]{\sin x}}{4-\sin^2 x}dx$
$$\int_0^\frac{\pi}{2} \frac{\sqrt[3]{\sin x}}{4-\sin^2 x}dx$$
I tried much of elementary methods to solve above integral but is not advancing.
Any methods from elementary to advanced are appreciated.
| Here is a way to simplify @Olivier Oloa's answer for the integral that is written in terms of the Gauss hypergeometric function.
Starting from (4) from a previous answer I gave here, it was shown that
$$_2F_1 \left (1, \frac{2}{3}; \frac{7}{6}; \frac{1}{4} \right ) = \frac{2^{4/3}}{\sqrt{\pi}} \Gamma \left (\frac{4}{3}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3135901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Solve $\frac{dx}{dt}=3x-5y$ $\frac{dy}{dt}=5x-3y$ Given that $x(0)=1$ and $y(0)=1$
Solve
$$\frac{dx}{dt}=3x-5y$$ $$\frac{dy}{dt}=5x-3y$$ Given that $x(0)=1$ and $y(0)=1$
So I've got to the stage where i have two general solutions and I now need the particular solutions, but I don't know how to find one of the vari... | There are several ways to solve this system. Here is yet another approach using eigenvalue/vector methods from linear algebra.
Let's re-write the system as:
$$ \frac{d\vec{r}}{dt} = \begin{pmatrix}3 &-5\\5 &-3 \end{pmatrix}\vec{r}(t) = \pmb{A}\,\vec{r}(t)$$
where $\vec{r}(t)=\begin{pmatrix}x(t)\\y(t)\end{pmatrix}$. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3137643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to use a conditional statement in an infinite series to approach a specific limit If I were to sum the infinite series:
$$\sum_{i=1}^\infty \frac{1}{2^n} = \frac{1}{2^1}+\frac{1}{2^2}...\frac{1}{2^\infty}$$
but as soon as the previous partial sum
$\sum_{i=1}^\infty \frac{1}{2^{n-1}}$ becomes greater than 0.9 I sub... | Claim:
The sign of the $k$-th term is precisely $(-1)^{\lfloor\tfrac{k+1}{2}\rfloor}$, excep for $k=1$ and $k=2$. To be precise, setting
$$f(1):=\frac12,
\qquad
f(2):=\frac14,
\qquad\text{ and }\qquad
f(k):=\frac{(-1)^{\lfloor\tfrac{k+1}{2}\rfloor}}{2^k}
\quad
\text{ for all }k>2,$$
yields the desired sum. This means ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3139360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solve the inequality $x^2 - 3 > 0$ For the inequality $x^2 - 3 > 0$, we have
\begin{align} x^2 - 3 & = (x+\sqrt 3)(x- \sqrt 3) > 0 \end{align}
Therefore,
\begin{align} x > -\sqrt 3 \end{align}
and
\begin{align} x > \sqrt 3 \end{align}
But, this is clearly wrong as we should get $x < -\sqrt 3$ and $x > \sqrt 3$ as the ... | HINT
Remember $ab > 0$ if $a,b > 0$ or $a,b < 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3139705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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Evaluate $\int \frac{x+4}{x^2 + 2x + 5}$ I am having issues with this integral. I am not sure if it is irreducible or not. I can't use the quadratic formula, but I can rearrange the integral to be $\int \frac{x+4} {(x^2 + 2x + 1) + 4}$, but I don't know how to deal with the $+4$.
Here is my work treating the quadratic... | The denominator has no real roots, which means you'll try to rewrite as follows ($A,B\in\mathbb{R}$):
$$\frac{x+4}{x^2 + 2x + 5} = A\underbrace{\frac{\left(x^2 + 2x + 5\right)'}{x^2 + 2x + 5}}_{\to \ln}+\underbrace{\frac{B}{x^2 + 2x + 5}}_{\to \arctan}$$
where $\left(x^2 + 2x + 5\right)'=2x+2$, so this comes down to fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3143709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Evaluating $\lim\limits_{x \to 0} \frac{(1+x)^{1/x} - e + \frac{1}{2}ex}{x^2}$ without expansions in limits
Evaluate $\lim\limits_{x \to 0} \frac{(1+x)^{1/x} - e + \frac{1}{2}ex}{x^2}$
One way that I can immediately think of is expanding each of the terms and solving like,
$$(1+x)^{1/x} = e^{\log_e (1+x)^{1/x}} = e^... | Solution without expansions by the L'Hospital's rule only:
$$\lim_{x\rightarrow0}\frac{(1+x)^{1/x} - e + \frac{1}{2}ex}{x^2}=\lim_{x\rightarrow0}\frac{(1+x)^{\frac{1}{x}}\left(\frac{\ln(1+x)}{x}\right)'+\frac{1}{2}e}{2x}=$$
$$=\lim_{x\rightarrow0}\frac{\frac{(1+x)^{\frac{1}{x}}\left(\frac{x}{1+x}-\ln(1+x)\right)}{x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3144560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Differentiate $11x^5 + x^4y + xy^5=18$ I am not sure how to differentiate $11x^5 + x^4y + xy^5=18$. I have a little bit of experience with implicit differentiation, but I'm not sure how to handle terms where both variables are multiplied together.
I have tried
$$\frac{d}{dx}(11x^5 + x^4y+xy^5) = \frac{d}{dx}(18)$$
$$\f... | differentiating each term
$$\frac{d}{dx} (11x^5)=11(5x^4) = 55x^4$$
$$\frac{d}{dx} (x^4y) = [4x^3 \cdot y] + [1 \cdot x^4\color{red}{\frac{dy}{dx}}] $$
$$\frac{d}{dx}(xy^5) = [1 \cdot y^5] + [x \cdot 5y^4\color{red}{\frac{dy}{dx}}] $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3147423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
} |
Solve the recurrence $b_1 = 2$ for $b_n = 3b_{n-1} + 5$
Solve the recurrence $b_1 = 2$ for $$b_n = 3b_{n-1} + 5$$
I've tried solving this problem using iteration, but the formula I get in the end is wrong. It is not a closed formula since there's still recurrence. I think my error is starts from the last line below. ... | Put $k =n-1$ and you get,
\begin{align}b_n &= 3^{n-1}b_1 + 3^{n-2} \cdot 5 + 3^{n-3} \cdot 5 + \dots +5 \\
&= 2 \cdot 3^{n-1} + 5(1 + 3 + 3^2 + \dots + 3^{n-2}) \\
&= 2\cdot 3^{n-1} + 5\left( \frac{3^{n-1}-1}{2}\right)\\
&= \frac{3^{n+1}-5}{2}
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3148196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
All the roots of $5\cos x - \sin x = 4$ in the interval $0^{\circ} \leq x \leq 360^{\circ}$? This is a problem that I stumbled upon in one of my books.
Representing $5\cos x - \sin x$ in the form $R\cos(x + \alpha)$ (as demanded by the question):
$
\rightarrow R = \sqrt{5^2 + ({-}1)^2} = \sqrt{26}\\
\rightarrow R\cos x... | This is just my way of saying what everyone else has already said.
\begin{align}
R &=\sqrt{1^2+5^2}=\sqrt{26} \\
\cos \alpha &= \dfrac{5}{\sqrt{26}} \\
\sin \alpha &= \dfrac{1}{\sqrt{26}} \\
\alpha &\approx 11.31^\circ \\
\hline
5\cos x - 1 \sin x &= 4 \\
\cos x \cos \alpha - \sin x \sin \alpha &= \d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3148923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
$e^{\frac{1}{x}} < 1 + \frac{1}{x-1} $ I want to prove that $e^{1/x} < 1 + \frac{1}{x-1}$ for $x > 1$.
The first thing I tried is differentiating $f(x) = e^{1/x} - 1 - \frac{1}{x-1}$: this gives
$$ \frac{1}{x^2} \left( \left(1 + \frac{1}{x-1}\right)^2 - e^{\frac{1}{x}} \right) $$
If I could show that $\left(1 + \frac{... | Prove instead that, for $0<x<1$,
$$
e^x<1+\frac{1}{\frac{1}{x}-1}=1+\frac{x}{1-x}=\frac{1}{1-x}
$$
which is the same as proving that, for $0<x<1$,
$$
e^{1-x}<\frac{1}{x}
$$
that is, $e^x>ex$. Now differentiating is easier, isn't it? If $f(x)=e^x-ex$,
$$
f'(x)=e^x-e
$$
Thus $f'$ is negative for $0<x<1$. Since $f(1)=0$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3149211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 1
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Find all polynomials such that $p(x^2-2x)=p(x-2)^2$
Find all polynomials $p\in \mathbb{C}[x]$ such that $$p(x^2-2x)=p(x-2)^2$$
We can not say anything specific about the degree since both sides are of the degree $2n$.
Also by copering the coeficients we see that the leading coefficent must be $1$.
Setting $$x^2-2x = ... | EDIT: (I made a change of variable $x-1\mapsto x$ as suggested by @Hw Chu.) We need to solve $\displaystyle p((x-1)^2-1)=p(x-1-1)^2$ or $p(x^2-1)=p(x-1)^2$ equivalently. Let $$\displaystyle q(x) =p(x-1)= cx^k\prod_{1\le i\le N\\a_i\ne 0} (x-a_i)^{n_i},$$ where $c\ne 0$ and $a_j\ne a_k$ for $j\ne k$. Plugging this into... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3149891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Finding $f(2019)$ in definite integration If $\displaystyle f(n)=\int^{1}_{0}(1+x+x^2+\cdots +x^{n-1})(1+3x+5x^2+\cdots +(2n-1)x^{n-1})dx$. Then $f(2019)$ is
What I tried:
$$1+x+x^2+\cdots +x^{n-1}=\frac{1-x^n}{1-x}$$
and $$1+3x+5x^2+\cdots +(2n-1)x^{n-1}=\frac{1}{1-x}+\frac{2(1-x^n)}{1-x}-\frac{(2n-1)x^n}{1-x}$$
How ... | By making change of variable $x=t^2$, we obtain
$$\begin{align*}
f(n)&=2\int_0^1 t(1+t^2+\cdots +t^{2n-2})(1+3t^2+5t^4+\cdots +(2n-1)t^{2n-2})dt.
\end{align*}$$ Further making substitution $u=t+t^3+\cdots +t^{2n-1}$, we get
$$
f(n)=2\int_0^n u\ du=\left[u^2\right]^n_0=n^2.
$$ So $f(2019)=2019^2$ follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3151253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Entropy of the upper and lower bits of a square number Consider a uniformly random number $x<2^n$. Let $H_{\star}(n)$ denote the (base-$2$ Shannon) entropy of the first $n$ bits of $x^2$, and let $H^{\star}(n)$ denote the entropy of the rest of the bits of $x^2$. In other words, writing $x^2 = q2^n + r$ with $r<2^n$, $... | With help from Peter Taylor's comment, I was able to prove the expression for $H_{\star}(n)$. I believe the approximation for $H^{\star}(n)$ could also be proven by noting that for $x>2^{n/2}$, $\sqrt{x^2-2^{n-1}}\leqslant \sqrt{q}$ or $\sqrt{q}\leqslant\sqrt{x^2+2^{n-1}}$, and then expanding these bounds as Taylor ser... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3152504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Limit of the sum using integral $\lim\limits_{n\rightarrow\infty}\sum_{k = 1}^{n} \frac{1}{(k+n)\sqrt{1 + n\ln({1+\frac{k}{n^2}})}}$. I can find it using integral: $\lim\limits_{n\rightarrow\infty}\frac{1}{n}\sum_{k = 1}^{n} \frac{1}{(1 + \frac{k}{n})\sqrt{1 + n\ln({1+\frac{k}{n^2}})}}$, but how get rid of $n$ and $\fr... | We have
\begin{align}
\frac 1{\sqrt{1 + n\ln\left({1+\frac{k}{n^2}}\right)}}
&=\left(1 + n\ln\left({1+\frac{k}{n^2}}\right)\right)^{-1/2}\\
&=\left(1 + \frac kn+O\left(\frac{k^2}{n^3}\right)\right)^{-1/2}\\
&=\frac 1{\sqrt{1 + \frac kn}}+\frac 1nO\left(\frac{k}{n}\right)^2
\end{align}
Consequently,
\begin{align}
\sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3153824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Yet another difficult logarithmic integral This question is a follow-up to MSE#3142989.
Two seemingly innocent hypergeometric series ($\phantom{}_3 F_2$)
$$ \sum_{n\geq 0}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2\frac{(-1)^n}{2n+1}\qquad \sum_{n\geq 0}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2\frac{1}{2^n(2n+1)}$$
can b... | Long Comment: Notes on evaluating $I_A$:
I first found the equivalent integral
$$I_A=2 \int_0^1 \frac{\sinh ^{-1}(x)}{x \sqrt{1-x^2}} \, dx\tag{1}$$
Integrating (1) by parts I then found
$$I_A=2 \int_0^1 \frac{\log \left(\frac{\sqrt{1-x^2}+1}{x}\right)}{\sqrt{x^2+1}} \, dx\tag{2}$$
Then since
$$\frac{1}{\sqrt{x^2+1}}=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3154337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 1,
"answer_id": 0
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integral $C(a,b)=\int_0^{2\pi}\frac{xdx}{a+b\cos^2 x}$ I am looking for other methods to find the general integral
$$C(a,b)=\int_0^{2\pi}\frac{xdx}{a+b\cos^2x}$$
To do so, I first preformed $u=x-\pi$:
$$C(a,b)=\int_{-\pi}^{\pi}\frac{xdx}{a+b\cos^2x}+\pi\int_{-\pi}^{\pi}\frac{dx}{a+b\cos^2x}$$
The sub $x\mapsto -x$ prov... | Note that $\cos^2(x)=\frac{1+\cos(2x)}{2}$. Therefore,
$$\begin{align}
C(a,b)&=2\pi\int_0^\pi \frac1{a+b\cos^2(x)}\,dx\\\\
&=2\pi \int_0^\pi \frac{2}{2a+b+b\cos(2x)}\,dx\\\\
&=2\pi \int_0^{2\pi}\frac{1}{2a+b+b\cos(x)}\,dx\tag1\\\\
&=4\pi \int_0^\pi \frac{1}{2a+b+b\cos(x)}\,dx\tag2
\end{align}$$
We can proceed by eith... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3154937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Computing $ \lim _{n \rightarrow \infty}\left[n-\frac{n}{e}\left(1+\frac{1}{n}\right)^{n}\right]$
$$
\lim _{n \rightarrow \infty}\left[n-\frac{n}{e}\left(1+\frac{1}{n}\right)^{n}\right] \text { equals }\_\_\_\_
$$
I tried to expand the term in power using binomial theorem but still could not obtain the limit.
| One has
$$n - \frac{n}{e} \left( 1 + \frac{1}{n}\right)^n = n - \frac{n}{e} \exp \left( n \ln \left( 1 + \frac{1}{n}\right)\right) =n - \frac{n}{e} \exp \left( n \left(\frac{1}{n}- \frac{1}{2n^2} + o \left( \frac{1}{n^2}\right)\right)\right) $$
so
$$n - \frac{n}{e} \left( 1 + \frac{1}{n}\right)^n = n - \frac{n}{e} \e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3155463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find the area of $y = 12-x^2, y=x^2-6$ I found the x-intercepts via my graphing calculator to be $(-3,3)$ and $(3,3)$ and from it I formed the Area equation,
$$A = \int_{-3}^{3} (12-x^2 - x^2 - 6)dx = \int_{-3}^{3}(-2x^2+6)dx \\
= -\frac{2}{3}x^3 + 6x \bigg]_{-3}^{3}$$
$$A = \bigg(3\frac{2}{3}\cdot 3^3 + 6(3) \bigg) ... | $$A = \int_{-3}^{3} (12-x^2 - (x^2 - 6))dx = \int_{-3}^{3}(-2x^2+18)dx
= -\frac{2}{3}x^3 + 18x \bigg]_{-3}^{3}$$
$$A = \bigg(-\frac{2}{3}\cdot 3^3 + 18(3) \bigg) - \bigg(-\frac{2}{3} \cdot -3^3 + 18(-3)\bigg)=36-(-36)=72.$$
You forgot the second set of parentheses inside the first integral. You don't need to split up ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3156121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Prove: if $c^2+8 \equiv 0$ mod $p$ then $c^3-7c^2-8c$ is a quadratic residue mod $p$. I want to show:
If $c^2+8 \equiv 0$ mod $p$ for prime $p>3$, then $c^3-7c^2-8c$ is a quadratic residue mod $p$.
I have calculated that $c^3-7c^2-8c \equiv -7c^2-16c \equiv 56- 16c \equiv 8(7-2c) \equiv c^2 (2c -7)$, so it should be ... | Hint:
As $c^2\equiv-8\pmod p,$
$$c(c+1)=c^2+c\equiv c-8\pmod p$$
$c(c+1)(c-8)\equiv?$
I believe this is how the problem naturally came into being .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3157905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
How can I block diagonalise this matrix? I have this matrix:
$$A = \left(
\begin{array}{cccc}
0 & 0 & 0 & -1 \\
1 & 0 & 0 & -1 \\
0 & 1 & 0 & -1 \\
0 & 0 & 1 & -1 \\
\end{array}
\right)$$
and I would like to show that it can be block diagonalised into :
$$ B = \left(
\begin{array}{cccc}
\cos 2\pi/5 & -\sin 2\pi/5 ... | it is already a companion matrix (both that form and its transpose are used, depends on circumstances). The characteristic polynomial is $$x^4 + x^3 + x^2 + x + 1 = \frac{x^5-1}{x-1}$$
With four distinct eigenvalues (complex) it diagonalizes. Next, you need to figure out how to take a specific diagonal matrix, with com... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3158856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Can someone shed some light on this inequality? I have a question relating to image that I've attached. It is a proof that the sequence is increasing. I don't understand the logic behind the third equation $$\frac{a_{n+1}}{a_n}>\left (1-\frac{1}{n+1}\right ) \left (\frac{n+1}{n}\right)$$
where does the equation in the ... | So, we have
$$\frac{a_{n+1}}{a_n} = \left(1 - \frac{1}{(n+1)^2}\right)^{n+1}\left(\frac{n+1}{n}\right).$$
The author then applies Bernoulli's inequality to the first term on the RHS:
$$\left(1 - \frac{1}{(n+1)^2}\right)^{n+1} > 1 + (n+1)\left(\frac{-1}{(n+1)^2}\right) = 1 - \frac{1}{n+1}.$$
We can now return to the fir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3165778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Solving $\int \frac{\sqrt{(x-2)^3}}{(\sqrt{x+1})^2} dx$ This is the last indefinite integral I am attempting before proceeding to definite integrals study. Any ideas how to solve it?
$$\int \frac{\sqrt{(x-2)^3}}{(\sqrt{x+1})^2} dx$$
My attempt: (not the longest one... :P)
$$\int \frac{\sqrt{(x-2)^3}}{(\sqrt{x+1})^2} dx... | Consider
$$I = \int \frac{\sqrt{(x-2)^3}}{(\sqrt{x+1})^2} dx = \int \frac{\sqrt{(x-2)^3}}{x+1} dx$$
Let $u = x -2 \implies du = dx $
$$I = \int\frac{u^\frac{3}{2}}{u + 3}du$$
Let $u = 3\tan^2t \implies du = 6\tan t \sec^2t dt$
$$I = 6\sqrt{27}\int\frac{\tan^4t\sec^2t}{3\sec^2t}dt$$
$$ = 6\sqrt{3}\int\tan^4t \ dt$$
$$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3167804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What is wrong with this solution of find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$ What is wrong with this solution of find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$
They all are positive terms so arithmetic mean is greater than equal to geometric mean.
$$ \sec^6 x +\csc^6 x + \sec^6... | Others explain why $48$ is correct as a a lower bound but may not be the sharp lower bound.
One way to get a lower bound of $80$ involves using the fact that each term is a cubed quantity. Start with the decomposition
$\sec^6 x + \csc^6 x + \sec^6 x\csc^6 x=A+B$
$A=\sec^6 x + \csc^6 x$
$B=\sec^6 x\csc^6 x$
Factor $A$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3168377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Simultaneous P.D.E. problem
Solve the following differential equation:
$$ \frac{dx}{y^2(x-y)} = \frac{dy}{x^2(x-y)} = \frac{dz}{z(x^2 + y^2)} $$
I have so far got
$x^2dx = y^2dy $ which implies that $(x-y)(x^2 +xy + y^2) = a$ ...(1)
Then $$\frac{dx+ dy}{(x-y)} = \frac{dz}{z}$$
Using the equation (1), I get
$\frac{... | $$ \frac{dx}{y^2(x-y)} = \frac{dy}{x^2(x-y)} = \frac{dz}{z(x^2 + y^2)} $$
This system of ODEs might come from solving the PDE :
$$y^2(x-y)\frac{\partial z}{\partial x}+x^2(x-y)\frac{\partial z}{\partial y}=(x^2+y^2)z$$
A first characteristic equation comes from $\frac{dx}{y^2(x-y)} = \frac{dy}{x^2(x-y)} $ which solutio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3171667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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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.