Question

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

score 682 · Answer 1 · 2026-05-07 10:14:09

682

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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 07 May 2026 - 10:14

score 451 · Answer 2 · 2026-04-25 12:42:38

451

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

Published on 25 Apr 2026 - 12:42

score 248 · Answer 3 · 2026-04-15 06:34:02

248

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

Published on 15 Apr 2026 - 6:34

score 105 · Answer 4 · 2026-04-16 16:06:07

105

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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 16 Apr 2026 - 16:06

score 314 · Answer 5 · 2026-04-16 05:06:46

314

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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 16 Apr 2026 - 5:06

score 474 · Answer 6 · 2026-04-16 07:25:26

474

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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 16 Apr 2026 - 7:25

score 395 · Answer 7 · 2026-04-16 23:59:19

395

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

Published on 16 Apr 2026 - 23:59

score 606 · Answer 8 · 2026-04-17 09:37:56

606

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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 17 Apr 2026 - 9:37

score 607 · Answer 9 · 2026-04-16 17:06:28

607

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

Published on 16 Apr 2026 - 17:06

score 406 · Answer 10 · 2026-04-15 15:34:45

406

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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 15 Apr 2026 - 15:34

score 55 · Answer 11 · 2026-04-16 09:46:52

55

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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 16 Apr 2026 - 9:46

score 107 · Answer 12 · 2026-05-10 15:30:04

107

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

Published on 10 May 2026 - 15:30

score 53 · Answer 13 · 2026-04-13 22:54:42

53

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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 13 Apr 2026 - 22:54

score 152 · Answer 14 · 2026-04-17 04:36:47

152

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

Published on 17 Apr 2026 - 4:36

score 305 · Answer 15 · 2026-04-15 04:15:41

305

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 15 Apr 2026 - 4:15

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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