User Ali Shadhar - Mathematics Stack Exchange

13 votes

Compute $\sum_{n=1}^\infty\frac{H_n^2H_n^{(2)}}{n^3}$

answered Aug 5, 2019 at 22:13

13 votes

Prove $\zeta(3)=2\sum_{n=1}^\infty\frac{H_n}{n}\left[\frac1{4^n}{2n\choose n}\left(H_{2n}-H_n-\frac1{2n}-\ln2\right)+\frac1{2n}\right]$

answered Nov 7, 2019 at 16:33

12 votes

Accepted

How can I prove $\frac{\gamma}{2}=\int_{0}^{\infty}\frac{e^{-x^{2}}-e^{-x}}{x}\text{d}x$?

answered Aug 2, 2019 at 22:04

11 votes

How to compute $\sum_{n=1}^\infty\frac{(-1)^nH_n}{n^4}$ by real integration only?

answered Nov 25, 2019 at 4:58

11 votes

Proving that $\int_0^1 \frac{\arctan x}{x}\ln\left(\frac{1+x^2}{(1-x)^2}\right)dx=\frac{\pi^3}{16}$

answered Apr 27, 2019 at 16:55

11 votes

Accepted

prove $\ln(1+x^2)\arctan x=-2\sum_{n=1}^\infty \frac{(-1)^n H_{2n}}{2n+1}x^{2n+1}$

answered Sep 29, 2019 at 4:03

11 votes

Challenging sum: Calculate $\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^52^k{2k \choose k}}$

answered Jul 30, 2019 at 9:07

10 votes

Evaluating $\int_0^{\pi/2} \frac{t \ln (1-\sin{t})}{\sin t} dt$

answered Nov 16, 2019 at 20:22

10 votes

Conjectural closed-form of $\int_0^1 \frac{\log^n (1-x) \log^{n-1} (1+x)}{1+x} dx$

answered Oct 30, 2019 at 18:48

10 votes

Mind-blowing Sums: Compute $\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^22^n}$ and $\sum_{n=1}^\infty\frac{H_n^3}{n^22^n}$

answered Aug 19, 2019 at 20:17

10 votes

Accepted

A suprising conjectural closed-form of $\sum _{n=1}^{\infty } \frac{1}{n^4 2^n \binom{3 n}{n}}$ and integral variations

answered Mar 31, 2020 at 8:46

10 votes

Accepted

$\zeta(4)$ in terms of a series of $\zeta(3)$ and harmonic numbers

answered Jul 12, 2020 at 5:08

10 votes

Tough definite integral: $\int_0^\frac{\pi}{2}x\ln^2(\sin x)~dx$

answered Aug 26, 2020 at 7:11

10 votes

Accepted

Is there an efficient way of showing $\int_{-1}^{1} \ln\left(\frac{2(1+\sqrt{1-x^2})}{1+x^2}\right)dx = 2$?

answered Aug 18, 2020 at 0:10

9 votes

Compute $\sum_{n=1}^\infty\frac{H_n^2H_n^{(2)}}{n^3}$

answered Aug 6, 2019 at 6:39

9 votes

Accepted

Compute $\int_0^\infty \frac{\operatorname{Li}_3(x)}{1+x^2}\ dx$

answered Aug 14, 2019 at 9:37

9 votes

Find the closed form of $\sum_{n=1}^{\infty} \frac{H_{ n}}{2^nn^4}$

answered Apr 20, 2019 at 19:46

9 votes

Two powerful alternating sums $\sum_{n=1}^\infty\frac{(-1)^nH_nH_n^{(2)}}{n^2}$ and $\sum_{n=1}^\infty\frac{(-1)^nH_n^3}{n^2}$

answered Jun 8, 2019 at 2:24

8 votes

Evaluating $\sum_{n=1}^\infty\frac{(H_n)^2}{n}\frac{\binom{2n}n}{4^n}$

answered Jun 7, 2019 at 5:43

8 votes

Sum of Harmonic numbers $\sum\limits_{n=1}^{\infty} \frac{H_n^{(2)}}{2^nn^2}$

answered Apr 25, 2019 at 23:33

8 votes

Accepted

Compute $\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^42^k{2k \choose k}}$

answered Jul 29, 2019 at 2:09

8 votes

A group of important generating functions involving harmonic number.

answered Sep 23, 2019 at 4:34

8 votes

Another beautiful integral (Part 2)

answered Jan 12, 2020 at 0:00

8 votes

Evaluating $\int_0^1\frac{\ln(x)\ln(1-x)\ln(1+\sqrt{1-x^2})}{x}\mathrm{d}x$

answered Jun 21, 2021 at 17:05

8 votes

Evaluating $\int_0^{\frac{\pi}{2}}x^2 \cot x\ln(1-\sin x)\mathrm{d}x$

answered Jan 26, 2023 at 6:52

8 votes

Conjecture for Integrals of the Form $\int_{0}^{1}\frac{\ln^n\left(1-x^{2}\right)}{1+x}dx$

answered Oct 21, 2023 at 7:08

7 votes

Show $\int_0^\infty \frac{\ln^2x}{(x+1)^2+1} \, dx=\frac{5\pi^3}{64}+\frac\pi{16}\ln^22$

answered Sep 25, 2020 at 16:09

7 votes

Accepted

Prove that $\sum_{n=2}^{\infty} \frac{(-1)^{n}}{n}\zeta(n) = \gamma$

answered Jan 30, 2023 at 5:13

7 votes

Accepted

How to evaluate $\int _0^{\frac{\pi }{2}}x\ln \left(\sin \left(x\right)\right)\:dx$

answered Jul 8, 2020 at 6:28

7 votes

A closed form for the dilogarithm integral $\int _{ 0 }^{ 1 }{ \frac { \operatorname{Li}_2\left( 2x\left( 1-x \right) \right) }{ x } dx } $

answered Jun 6, 2020 at 5:51

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498 Answers

Compute $\sum_{n=1}^\infty\frac{H_n^2H_n^{(2)}}{n^3}$

Prove $\zeta(3)=2\sum_{n=1}^\infty\frac{H_n}{n}\left[\frac1{4^n}{2n\choose n}\left(H_{2n}-H_n-\frac1{2n}-\ln2\right)+\frac1{2n}\right]$

How can I prove $\frac{\gamma}{2}=\int_{0}^{\infty}\frac{e^{-x^{2}}-e^{-x}}{x}\text{d}x$?

How to compute $\sum_{n=1}^\infty\frac{(-1)^nH_n}{n^4}$ by real integration only?

Proving that $\int_0^1 \frac{\arctan x}{x}\ln\left(\frac{1+x^2}{(1-x)^2}\right)dx=\frac{\pi^3}{16}$

prove $\ln(1+x^2)\arctan x=-2\sum_{n=1}^\infty \frac{(-1)^n H_{2n}}{2n+1}x^{2n+1}$

Challenging sum: Calculate $\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^52^k{2k \choose k}}$

Evaluating $\int_0^{\pi/2} \frac{t \ln (1-\sin{t})}{\sin t} dt$

Conjectural closed-form of $\int_0^1 \frac{\log^n (1-x) \log^{n-1} (1+x)}{1+x} dx$

Mind-blowing Sums: Compute $\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^22^n}$ and $\sum_{n=1}^\infty\frac{H_n^3}{n^22^n}$

A suprising conjectural closed-form of $\sum _{n=1}^{\infty } \frac{1}{n^4 2^n \binom{3 n}{n}}$ and integral variations

$\zeta(4)$ in terms of a series of $\zeta(3)$ and harmonic numbers

Tough definite integral: $\int_0^\frac{\pi}{2}x\ln^2(\sin x)~dx$

Is there an efficient way of showing $\int_{-1}^{1} \ln\left(\frac{2(1+\sqrt{1-x^2})}{1+x^2}\right)dx = 2$?

Compute $\sum_{n=1}^\infty\frac{H_n^2H_n^{(2)}}{n^3}$

Compute $\int_0^\infty \frac{\operatorname{Li}_3(x)}{1+x^2}\ dx$

Find the closed form of $\sum_{n=1}^{\infty} \frac{H_{ n}}{2^nn^4}$

Two powerful alternating sums $\sum_{n=1}^\infty\frac{(-1)^nH_nH_n^{(2)}}{n^2}$ and $\sum_{n=1}^\infty\frac{(-1)^nH_n^3}{n^2}$

Evaluating $\sum_{n=1}^\infty\frac{(H_n)^2}{n}\frac{\binom{2n}n}{4^n}$

Sum of Harmonic numbers $\sum\limits_{n=1}^{\infty} \frac{H_n^{(2)}}{2^nn^2}$

Compute $\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^42^k{2k \choose k}}$

A group of important generating functions involving harmonic number.

Another beautiful integral (Part 2)

Evaluating $\int_0^1\frac{\ln(x)\ln(1-x)\ln(1+\sqrt{1-x^2})}{x}\mathrm{d}x$

Evaluating $\int_0^{\frac{\pi}{2}}x^2 \cot x\ln(1-\sin x)\mathrm{d}x$

Conjecture for Integrals of the Form $\int_{0}^{1}\frac{\ln^n\left(1-x^{2}\right)}{1+x}dx$

Show $\int_0^\infty \frac{\ln^2x}{(x+1)^2+1} \, dx=\frac{5\pi^3}{64}+\frac\pi{16}\ln^22$

Prove that $\sum_{n=2}^{\infty} \frac{(-1)^{n}}{n}\zeta(n) = \gamma$

How to evaluate $\int _0^{\frac{\pi }{2}}x\ln \left(\sin \left(x\right)\right)\:dx$

A closed form for the dilogarithm integral $\int _{ 0 }^{ 1 }{ \frac { \operatorname{Li}_2\left( 2x\left( 1-x \right) \right) }{ x } dx } $