Question

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

score 112 · Answer 1 · 2018-10-11 21:44:42

112

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

Published on 11 Oct 2018 - 21:44

score 193 · Answer 2 · 2018-10-14 16:12:10

193

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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 14 Oct 2018 - 16:12

score 157 · Answer 3 · 2018-10-14 17:18:52

157

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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 14 Oct 2018 - 17:18

score 235 · Answer 4 · 2018-11-19 03:55:06

235

Views

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 19 Nov 2018 - 3:55

score 74 · Answer 5 · 2018-11-21 15:15:39

74

Views

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 21 Nov 2018 - 15:15

score 141 · Answer 6 · 2018-11-25 07:18:01

141

Views

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

Published on 25 Nov 2018 - 7:18

score 294 · Answer 7 · 2018-11-25 19:12:50

294

Views

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

Published on 25 Nov 2018 - 19:12

score 164 · Answer 8 · 2018-12-04 08:28:41

164

Views

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

Published on 04 Dec 2018 - 8:28

score 259 · Answer 9 · 2018-12-07 11:17:37

259

Views

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

Published on 07 Dec 2018 - 11:17

score 1k · Answer 10 · 2018-12-14 21:01:14

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 14 Dec 2018 - 21:01

score 353 · Answer 11 · 2018-12-21 11:50:56

353

Views

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

Published on 21 Dec 2018 - 11:50

score 907 · Answer 12 · 2018-12-25 22:42:12

907

Views

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

Published on 25 Dec 2018 - 22:42

score 253 · Answer 13 · 2019-01-06 01:09:05

253

Views

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

Published on 06 Jan 2019 - 1:09

score 95 · Answer 14 · 2019-01-07 09:12:56

95

Views

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

Published on 07 Jan 2019 - 9:12

score 128 · Answer 15 · 2019-01-08 17:39:35

128

Views

Polylogarithmic integrals

Published on 08 Jan 2019 - 17:39

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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