Question

Evaluating $\sum_{n=1}^{\infty} \frac{\left(\frac{1}{2}\right)^n}{n^2}$

score 115 · Answer 1 · 2026-04-16 08:17:00

115

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Evaluating $\sum_{n=1}^{\infty} \frac{\left(\frac{1}{2}\right)^n}{n^2}$

Published on 16 Apr 2026 - 8:17

score 196 · Answer 2 · 2026-04-17 10:23:19

196

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Evaluating $\int_0^1\frac{\ln(x)}{1-x/2}\int_0^{x/2}\frac{\ln(1-t)}{t}\,dt\,dx$

Published on 17 Apr 2026 - 10:23

score 160 · Answer 3 · 2026-04-16 13:45:05

160

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

Published on 16 Apr 2026 - 13:45

score 237 · Answer 4 · 2026-04-16 06:09:18

237

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Improper integral $\int_0^\infty \frac{x^{\alpha}\ln x}{x^2+1}\,dx=\frac{\pi^2}{4} \frac{\sin(\pi \alpha/2)}{\cos^2(\pi \alpha/2)}$

Published on 16 Apr 2026 - 6:09

score 76 · Answer 5 · 2026-04-16 07:39:08

76

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Integrate $\int_{-\infty}^\infty [4(\log r_1 - \log r_2) - 2(x_1^2/r_1^2 - x_2^2/r_2^2)]^2 dx$

Published on 16 Apr 2026 - 7:39

score 144 · Answer 6 · 2026-04-14 02:43:56

144

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What is the value of this expression involving the dilogarithm $\operatorname{Li}_2$ and $\sqrt{2}$?

Published on 14 Apr 2026 - 2:43

score 297 · Answer 7 · 2026-04-16 15:11:19

297

Views

Closed form of $\int{\lfloor{x}\rfloor}dx$

Published on 16 Apr 2026 - 15:11

score 167 · Answer 8 · 2026-04-15 13:03:08

167

Views

Are there any non-trivial special values of $\operatorname{Li}_4(z)$?

Published on 15 Apr 2026 - 13:03

score 262 · Answer 9 · 2026-04-16 02:52:14

262

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Proving $\Im\operatorname{Li}_2(\sqrt i(\sqrt 2-1))=\frac34G+\frac18\pi\ln(\sqrt2-1)$

Published on 16 Apr 2026 - 2:52

score 1k · Answer 10 · 2026-04-16 20:00:15

1k

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Evaluate$\int\limits_0^1 [\log(x)\log(1-x)+\operatorname{Li}_2(x)]\left[\frac{\operatorname{Li}_2(x)}{x(1-x)}-\frac{\zeta(2)}{1-x}\right]\mathrm dx$

Published on 16 Apr 2026 - 20:00

score 356 · Answer 11 · 2026-04-15 14:24:28

356

Views

Upper bound the Polylogarithm $\sum_{n=1}^\infty \frac{x^n}{n^2}$

Published on 15 Apr 2026 - 14:24

score 910 · Answer 12 · 2026-04-17 05:56:09

910

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How to prove $\int_0^1x\ln^2(1+x)\ln(\frac{x^2}{1+x})\frac{dx}{1+x^2}$

Published on 17 Apr 2026 - 5:56

score 256 · Answer 13 · 2026-04-16 22:31:01

256

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Logarithmic integral $ \int_0^1 \frac{x}{x^2+1} \, \log(x)\log(x+1) \, {\rm d}x $

Published on 16 Apr 2026 - 22:31

score 98 · Answer 14 · 2026-04-15 00:36:07

98

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Polylog property $\operatorname{Li}_n(1-i)+(-1)^n\operatorname{Li}_n(\frac{1+i}2)$

Published on 15 Apr 2026 - 0:36

score 133 · Answer 15 · 2026-04-12 06:36:37

133

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Polylogarithmic integrals

Published on 12 Apr 2026 - 6:36

List Question

Evaluating $\sum_{n=1}^{\infty} \frac{\left(\frac{1}{2}\right)^n}{n^2}$

Evaluating $\int_0^1\frac{\ln(x)}{1-x/2}\int_0^{x/2}\frac{\ln(1-t)}{t}\,dt\,dx$

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

Improper integral $\int_0^\infty \frac{x^{\alpha}\ln x}{x^2+1}\,dx=\frac{\pi^2}{4} \frac{\sin(\pi \alpha/2)}{\cos^2(\pi \alpha/2)}$

Integrate $\int_{-\infty}^\infty [4(\log r_1 - \log r_2) - 2(x_1^2/r_1^2 - x_2^2/r_2^2)]^2 dx$

What is the value of this expression involving the dilogarithm $\operatorname{Li}_2$ and $\sqrt{2}$?

Closed form of $\int{\lfloor{x}\rfloor}dx$

Are there any non-trivial special values of $\operatorname{Li}_4(z)$?

Proving $\Im\operatorname{Li}_2(\sqrt i(\sqrt 2-1))=\frac34G+\frac18\pi\ln(\sqrt2-1)$

Evaluate$\int\limits_0^1 [\log(x)\log(1-x)+\operatorname{Li}_2(x)]\left[\frac{\operatorname{Li}_2(x)}{x(1-x)}-\frac{\zeta(2)}{1-x}\right]\mathrm dx$

Upper bound the Polylogarithm $\sum_{n=1}^\infty \frac{x^n}{n^2}$

How to prove $\int_0^1x\ln^2(1+x)\ln(\frac{x^2}{1+x})\frac{dx}{1+x^2}$

Logarithmic integral $ \int_0^1 \frac{x}{x^2+1} \, \log(x)\log(x+1) \, {\rm d}x $

Polylog property $\operatorname{Li}_n(1-i)+(-1)^n\operatorname{Li}_n(\frac{1+i}2)$

Polylogarithmic integrals

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