User user170231 - Mathematics Stack Exchange

32 votes

3 answers

2k views

Does $\int_{-1}^1\frac{\arctan x}{\text{arctanh}\,x}\,\mathrm{d}x$ have a closed form?

asked Jun 2, 2015 at 19:25

17 votes

5 answers

3k views

Why doesn't using the approximation $\sin x\approx x$ near $0$ work for computing this limit?

asked Jun 25, 2015 at 2:34

17 votes

1 answer

807 views

Using complex analysis, show that $\int_0^\infty\frac{\arctan(x)}{1+x^2}\,dx=\frac{\pi^2}8$

asked Oct 25, 2022 at 17:56

12 votes

1 answer

557 views

Using complex analysis, prove $\int\limits_0^\infty \frac{\log(x) \arctan(x)}{1 + x^2} \, dx = \frac78 \zeta(3)$

asked Nov 27, 2022 at 8:59

10 votes

1 answer

145 views

Are there certain conditions that $a_n$ must meet in order for this series to converge?

asked Sep 25, 2015 at 21:27

9 votes

2 answers

435 views

Computing $\sum\limits_{n=1}^\infty\frac{\sin n}{n}$ with residues

asked Dec 29, 2015 at 21:14

9 votes

1 answer

1k views

Why does this double infinite sum $\sum_{n=1}^\infty \sum_{k=n}^\infty\frac{1}{k!}$ converge to $e$?

asked Jan 13, 2016 at 23:28

9 votes

3 answers

360 views

Integrate $\ln^3(\sin(x))$ using Fourier series

asked Jul 14, 2022 at 19:22

8 votes

1 answer

340 views

Why is $\lim\limits_{n\to\infty} \prod\limits_{k=1}^n \left(1 + \frac{k+1}{n^2}\right) = \sqrt e$?

asked Mar 31, 2022 at 18:32

8 votes

4 answers

395 views

Show $\sum\limits_{k=0}^\infty (-1)^k \frac{\Gamma\left(k+\frac14\right)}{\Gamma\left(k+\frac74\right)} = -\Gamma\left(-\frac12\right) = 2\sqrt\pi$

asked Aug 3, 2022 at 20:10

8 votes

2 answers

3k views

How to visualize the real projective plane $\mathbb RP^2$ in three dimensions, if possible?

asked Apr 19, 2018 at 20:35

7 votes

2 answers

423 views

Use Fourier series of odd function to evaluate $1-\frac15+\frac17-\frac1{11}+\cdots$

asked May 8, 2017 at 18:17

7 votes

1 answer

1k views

Nonlinear 1st order ODE involving a rational function

asked Sep 22, 2014 at 18:01

7 votes

0 answers

822 views

How to integrate $\int_0^\infty\frac{dx}{1+x^n}$ [duplicate]

asked Jan 11, 2015 at 1:45

7 votes

3 answers

226 views

Proving the identity $\tan x+2\tan2x+4\tan4x+8\cot8x=\cot x$ by considering a more general form

asked Feb 20, 2019 at 22:52

6 votes

1 answer

225 views

Question about Meijer-G definition and identity

asked Aug 14, 2015 at 16:07

5 votes

0 answers

117 views

Given $x_{n+1}=x_n+\frac{c}{x_{n-1}}$, find the product of two successive terms $x_kx_{k+1}$

asked Dec 16, 2016 at 22:51

5 votes

5 answers

2k views

Does this double sum $\sum_{m=0}^\infty \sum_{n=0}^\infty \frac{m+n+mn}{2^m(2^m+2^n)}$ converge?

asked Aug 7, 2015 at 17:16

5 votes

5 answers

272 views

Using Fourier series to compute $1-\frac1{3^2}-\frac1{5^2}+\frac1{7^2}+\cdots$

asked Apr 7, 2022 at 22:48

5 votes

0 answers

1k views

"summand" is to "sum" $\left(\sum\right)$ as ___ is to "product" $\left(\prod\right)$

asked May 4, 2022 at 17:09

5 votes

5 answers

176 views

Show that $\cos^220^\circ-\cos20^\circ\sin10^\circ+\sin^210^\circ=\frac34$

asked Feb 18, 2019 at 19:09

4 votes

3 answers

251 views

On the log-cosine integral $\int\limits_0^{\pi/2}\ln(\cos(x))\,dx$

asked Mar 23, 2022 at 22:56

4 votes

3 answers

200 views

Permutations of alphabet with X following Y and vowels in alphabetical order

asked Feb 4, 2022 at 20:42

4 votes

0 answers

62 views

Predicting shape of regions defined by $f(x)^2+f(y)^2\le1$

asked Apr 6, 2023 at 18:02

4 votes

2 answers

78 views

If $\lim\limits_{x\to\alpha}\frac{x-2}{x^3-2x+m}=-\infty$, then what are the possible values for $\alpha$ and $m$?

asked Dec 29, 2018 at 0:14

4 votes

2 answers

197 views

Computing the definite integral $\int_0^1(-1)^{\left\lfloor\frac1x\right\rfloor}\,\mathrm dx$

asked Dec 30, 2020 at 20:55

4 votes

2 answers

146 views

Euler substitution inconsistency in evaluating $\int_{-1}^1 \frac{\mathrm dx}{\sqrt{1-x}+2+\sqrt{1+x}}$

asked Sep 29, 2021 at 17:51

4 votes

2 answers

185 views

Generalizing $\sum_{n=1}^\infty \frac{\Gamma(n)\Gamma(x)}{\Gamma(n+x)} = \frac1{x-1}$ to double sum

asked Sep 26, 2022 at 18:20

4 votes

2 answers

234 views

If $\prod\limits_{k=0}^5(5^{2^k}+6^{2^k})=6^x-5^y$, what is the value of $x-y$?

asked Apr 23, 2015 at 2:19

4 votes

3 answers

146 views

What is the best way to explain setting a restriction on $\delta$ in $\epsilon$-$\delta$ proofs?

asked May 15, 2015 at 16:14

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118 Questions

Does $\int_{-1}^1\frac{\arctan x}{\text{arctanh}\,x}\,\mathrm{d}x$ have a closed form?

Why doesn't using the approximation $\sin x\approx x$ near $0$ work for computing this limit?

Using complex analysis, show that $\int_0^\infty\frac{\arctan(x)}{1+x^2}\,dx=\frac{\pi^2}8$

Using complex analysis, prove $\int\limits_0^\infty \frac{\log(x) \arctan(x)}{1 + x^2} \, dx = \frac78 \zeta(3)$

Are there certain conditions that $a_n$ must meet in order for this series to converge?

Computing $\sum\limits_{n=1}^\infty\frac{\sin n}{n}$ with residues

Why does this double infinite sum $\sum_{n=1}^\infty \sum_{k=n}^\infty\frac{1}{k!}$ converge to $e$?

Integrate $\ln^3(\sin(x))$ using Fourier series

Why is $\lim\limits_{n\to\infty} \prod\limits_{k=1}^n \left(1 + \frac{k+1}{n^2}\right) = \sqrt e$?

Show $\sum\limits_{k=0}^\infty (-1)^k \frac{\Gamma\left(k+\frac14\right)}{\Gamma\left(k+\frac74\right)} = -\Gamma\left(-\frac12\right) = 2\sqrt\pi$

How to visualize the real projective plane $\mathbb RP^2$ in three dimensions, if possible?

Use Fourier series of odd function to evaluate $1-\frac15+\frac17-\frac1{11}+\cdots$

Nonlinear 1st order ODE involving a rational function

How to integrate $\int_0^\infty\frac{dx}{1+x^n}$ [duplicate]

Proving the identity $\tan x+2\tan2x+4\tan4x+8\cot8x=\cot x$ by considering a more general form

Question about Meijer-G definition and identity

Given $x_{n+1}=x_n+\frac{c}{x_{n-1}}$, find the product of two successive terms $x_kx_{k+1}$

Does this double sum $\sum_{m=0}^\infty \sum_{n=0}^\infty \frac{m+n+mn}{2^m(2^m+2^n)}$ converge?

Using Fourier series to compute $1-\frac1{3^2}-\frac1{5^2}+\frac1{7^2}+\cdots$

"summand" is to "sum" $\left(\sum\right)$ as ___ is to "product" $\left(\prod\right)$

Show that $\cos^220^\circ-\cos20^\circ\sin10^\circ+\sin^210^\circ=\frac34$

On the log-cosine integral $\int\limits_0^{\pi/2}\ln(\cos(x))\,dx$

Permutations of alphabet with X following Y and vowels in alphabetical order

Predicting shape of regions defined by $f(x)^2+f(y)^2\le1$

If $\lim\limits_{x\to\alpha}\frac{x-2}{x^3-2x+m}=-\infty$, then what are the possible values for $\alpha$ and $m$?

Computing the definite integral $\int_0^1(-1)^{\left\lfloor\frac1x\right\rfloor}\,\mathrm dx$

Euler substitution inconsistency in evaluating $\int_{-1}^1 \frac{\mathrm dx}{\sqrt{1-x}+2+\sqrt{1+x}}$

Generalizing $\sum_{n=1}^\infty \frac{\Gamma(n)\Gamma(x)}{\Gamma(n+x)} = \frac1{x-1}$ to double sum

If $\prod\limits_{k=0}^5(5^{2^k}+6^{2^k})=6^x-5^y$, what is the value of $x-y$?

What is the best way to explain setting a restriction on $\delta$ in $\epsilon$-$\delta$ proofs?