Question

How to find $\sum_{n=1}^{\infty}\frac{H_nH_{2n}}{n^2}$ using real analysis and in an elegant way?

score 678 · Answer 1 · 2026-03-25 01:20:40

678

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How to find $\sum_{n=1}^{\infty}\frac{H_nH_{2n}}{n^2}$ using real analysis and in an elegant way?

Published on 25 Mar 2026 - 1:20

score 444 · Answer 2 · 2026-03-19 13:42:50

444

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Compute this following integral without Fourier series : $\int_0^{\pi/4}x\ln(\tan x)dx$

Published on 19 Mar 2026 - 13:42

score 245 · Answer 3 · 2019-04-29 23:13:28

245

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Compute in closed form $\int_0^1\frac{\arctan{ax}}{\sqrt{1-x^{2}}}dx$

Published on 29 Apr 2019 - 23:13

score 102 · Answer 4 · 2019-04-30 21:30:05

102

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Compute in closed form : $\int_0^{\frac{π}{4}} x\ln(\tan x)\left(1-\frac{1}{\cos^2 x}\right)dx$

Published on 30 Apr 2019 - 21:30

score 311 · Answer 5 · 2019-05-04 06:41:48

311

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How can I compute this integral in closed form : $\int_0^{\frac{π}{4}}\ln^2(\tan x)dx$

Published on 04 May 2019 - 6:41

score 471 · Answer 6 · 2019-05-10 16:04:40

471

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A twisted hypergeometric series $\sum_{n=1}^\infty\frac{H_n}{n}\left(\frac{(2n)!}{4^n(n!)^2}\right)^2$

Published on 10 May 2019 - 16:04

score 392 · Answer 7 · 2019-05-14 19:08:57

392

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Integral: $\int_0^1\frac{\mathrm{Li}_2(x^2)}{\sqrt{1-x^2}}dx$

Published on 14 May 2019 - 19:08

score 603 · Answer 8 · 2019-05-20 16:35:27

603

Views

More on the integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$

Published on 20 May 2019 - 16:35

score 604 · Answer 9 · 2019-05-20 18:01:43

604

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Closed-forms for the integral $\int_0^1\frac{\operatorname{Li}_n(x)}{1+x}dx$?

Published on 20 May 2019 - 18:01

score 404 · Answer 10 · 2019-05-21 05:21:28

404

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More on the log sine integral $\int_0^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\mathrm{d}\theta$

Published on 21 May 2019 - 5:21

score 52 · Answer 11 · 2019-05-21 13:53:35

52

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Patterns for the polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ and Nielsen polylogarithm $S_{n,p}\big(\tfrac12\big)$?

Published on 21 May 2019 - 13:53

score 103 · Answer 12 · 2019-05-28 20:48:31

103

Views

How to solve $\int_0^\infty \frac{\pi^2 e^{-x}}{6x}-\frac{ \operatorname{Li}_2(e^{-x})}{1-e^{-x}}dx$?

Published on 28 May 2019 - 20:48

score 52 · Answer 13 · 2026-03-25 04:59:35

52

Views

How to derive the Dilogarithm reflection $\operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)=\frac{\pi^2}{6}-\ln(x)\ln(1-x)$?

Published on 25 Mar 2026 - 4:59

score 149 · Answer 14 · 2019-05-31 01:37:36

149

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Questions related to formulas for $f_k(s)=\left(k^{1-s}-1\right)\zeta (s)$

Published on 31 May 2019 - 1:37

score 302 · Answer 15 · 2019-05-31 06:12:56

302

Views

On generalizing the harmonic sum $\sum_{n=1}^{\infty}\frac{H_n}{n^k}z^n = S_{k-1,2}(1)+\zeta(k+1)$ when $z=1$?

Published on 31 May 2019 - 6:12

List Question

How to find $\sum_{n=1}^{\infty}\frac{H_nH_{2n}}{n^2}$ using real analysis and in an elegant way?

Compute this following integral without Fourier series : $\int_0^{\pi/4}x\ln(\tan x)dx$

Compute in closed form $\int_0^1\frac{\arctan{ax}}{\sqrt{1-x^{2}}}dx$

Compute in closed form : $\int_0^{\frac{π}{4}} x\ln(\tan x)\left(1-\frac{1}{\cos^2 x}\right)dx$

How can I compute this integral in closed form : $\int_0^{\frac{π}{4}}\ln^2(\tan x)dx$

A twisted hypergeometric series $\sum_{n=1}^\infty\frac{H_n}{n}\left(\frac{(2n)!}{4^n(n!)^2}\right)^2$

Integral: $\int_0^1\frac{\mathrm{Li}_2(x^2)}{\sqrt{1-x^2}}dx$

More on the integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$

Closed-forms for the integral $\int_0^1\frac{\operatorname{Li}_n(x)}{1+x}dx$?

More on the log sine integral $\int_0^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\mathrm{d}\theta$

Patterns for the polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ and Nielsen polylogarithm $S_{n,p}\big(\tfrac12\big)$?

How to solve $\int_0^\infty \frac{\pi^2 e^{-x}}{6x}-\frac{ \operatorname{Li}_2(e^{-x})}{1-e^{-x}}dx$?

How to derive the Dilogarithm reflection $\operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)=\frac{\pi^2}{6}-\ln(x)\ln(1-x)$?

Questions related to formulas for $f_k(s)=\left(k^{1-s}-1\right)\zeta (s)$

On generalizing the harmonic sum $\sum_{n=1}^{\infty}\frac{H_n}{n^k}z^n = S_{k-1,2}(1)+\zeta(k+1)$ when $z=1$?

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