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Faoler's user avatar
Faoler
  • Member for 1 year, 11 months
  • Last seen this week
  • Iraq
14 votes
Accepted

Evalaute $\sum_{k=1}^\infty \arctan \left(\frac{1}{1+3k+k^2}\right)$

6 votes
Accepted

Challenge: $\int_0^\infty\frac1{1+x}\ln\left(\prod_{k=0}^\infty(1+e^{-(2k+1)\sqrt{x}})(1+e^{-(2k+1)\pi^2\sqrt{x}})\right)\,dx=\pi^3/24+\pi/24-\ln^22$

6 votes

Prove the zeta log gamma integral $\int_0^1{\zeta(1-n,1-t)\ln\Gamma{(t)}}dt = \frac{H_n\zeta(-n)+\zeta’(-n)}{n}$

5 votes
Accepted

Integrating $\int_0^\infty \frac{x\sin(tx)}{1+x^2} dx$ without contour integration?

5 votes
Accepted

Evaluate the trigonometric integral $\int_0^{\pi/2} \frac{ \sqrt[3]{\tan x} }{ \left( \sin x + \cos x \right)^2 }\, dx$

4 votes

On the relationship between $\Re\operatorname{Li}_n(1+i)$ and $\operatorname{Li}_n(1/2)$ when $n\ge5$

3 votes
Accepted

Compute ${\sum_{n=1}^{+\infty}(-1)^n\left(\sum_{k=1}^{n}\frac{1}{2k-1}-\frac{ \ln n}{2}-\frac{\gamma}{2}-\ln 2\right)}$

3 votes

Evaluate $\prod_{n\geq2}\left(\frac{4}{e^2}\left(1+\frac{1}{n}\right)^{2n+1}\frac{n^2-1}{4n^2-1}\right)$

3 votes

On $\int_0^1\prod_{k=1}^\infty (1-x^k)\, dx$

3 votes

$\lim_{x \rightarrow 0} \frac{x -\sin x}{x (1 - \cos x)}$

2 votes

Convergence or divergence of the sum series with $\sum_{n\ge 1} \frac{(1-2^{1/n})^6}{(n+3)^p}$, $p\in \Bbb R$

2 votes
Accepted

Why is $\int \frac1{\cosh x} dx=2\arctan(\tanh(\tfrac x2))+C$?

2 votes
Accepted

What's wrong in this expansion of this erfc-like function?

2 votes

Does this Hypergeometric Function a Closed form?

2 votes
Accepted

How to evaluate $I = \int_{0}^{1}\frac{\arctan^2(x)\log(1-x)}{1+x^2}\,dx$

2 votes
Accepted

An integral Wolfram Alpha cannot take: $\displaystyle\int_{-1}^{+1}(-1)^{i\pi e^{-x\pi}}dx$

1 vote

Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$

1 vote
Accepted

$\lim_{{n \to \infty}} \left( \frac{1}{n} + \frac{1}{\sqrt{n^2 + n}} + \frac{1}{\sqrt{n^2 + 2n}} + \ldots + \frac{1}{\sqrt{n^2 + (n-1)n}} \right)$

1 vote
Accepted

Calculate $\sum\limits_{n = - \infty }^\infty {\frac{{\log \left( {{{\left( {n + \frac{1}{3}} \right)}^2}} \right)}}{{n + \frac{1}{3}}}} $

1 vote
Accepted

How did Rudin change the order of the double sum $\sum_{n=0}^\infty c_n\sum_{m=0}^n\binom nma^{n-m}(x-a)^m$?

1 vote

How to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n (x^p+1)^p$?

1 vote

how to find expression of $a\text{ mod } n$ using polynomial of $e^{\frac{2a\pi i}{n}}$?

0 votes

$13\mid4^{2n+1}+3^{n+2}$

-1 votes

Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \,dx$