Question

evaluate the summation : $\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+1)(n+2x+3)}$

score 169 · Answer 1 · 2019-07-15 09:35:09

169

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evaluate the summation : $\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+1)(n+2x+3)}$

Published on 15 Jul 2019 - 9:35

score 180 · Answer 2 · 2026-03-25 16:01:56

180

Views

Generalizing Oksana's trilogarithm relation to $\text{Li}_3(\frac{n}8)$?

Published on 25 Mar 2026 - 16:01

score 1.4k · Answer 3 · 2019-07-15 20:54:20

1.4k

Views

Is there a closed-form solution for $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(3n+m)}$?

Published on 15 Jul 2019 - 20:54

score 242 · Answer 4 · 2019-07-15 23:49:33

242

Views

Find triple summation rel in a closed form $S=\sum_{n=1}^{\infty}\sum_{m=1}^{n}\sum_{k=1}^{m}\frac{1}{(n+1)(k+1)(m+1)nmk}$

Published on 15 Jul 2019 - 23:49

score 71 · Answer 5 · 2019-07-16 16:24:18

71

Views

some identity of summation and generalization

Published on 16 Jul 2019 - 16:24

score 313 · Answer 6 · 2019-07-18 01:26:22

313

Views

Evaluate $\int_0^{\frac{\pi}{4}}\frac{1+\cos x-x^{2}}{(1+x\sin x)\sqrt {1-x^{2}}}\,dx$

Published on 18 Jul 2019 - 1:26

score 191 · Answer 7 · 2019-07-18 03:06:53

191

Views

Evaluate the integral : $\int_{0}^{\infty}e^{-2x}Ei^{2}(x)dx$

Published on 18 Jul 2019 - 3:06

score 170 · Answer 8 · 2026-03-25 14:25:13

170

Views

Expressing $\sum_{n = 1}^\infty \sum_{k = 1}^n \frac{1}{n^4 k\,2^k}$ as a finite sum involving $\zeta(\cdot)$, $Li_k(\cdot)$, $\pi$, and $\ln 2$

Published on 25 Mar 2026 - 14:25

score 356 · Answer 9 · 2019-07-19 12:01:22

356

Views

Prove ${_4 F_3} \left(\frac12, \frac12, 1, 1; \frac34, \frac54, \frac32; \frac14 \right)= \frac14 \left(\frac{\pi^2}{4}+\log^2 (2+\sqrt{3} ) \right)$

Published on 19 Jul 2019 - 12:01

score 379 · Answer 10 · 2026-03-26 04:36:35

379

Views

Closed form for $\sum _{k=0}^n (-1)^k \binom{n}{k} \binom{n+k}{n} (H_n-H_k) x^k$

Published on 26 Mar 2026 - 4:36

score 361 · Answer 11 · 2019-07-21 05:42:56

361

Views

Finding a closed form for recurrence relations $a_n=na_{n-1}+1$ and $a_n=na_{n-1}+n$

Published on 21 Jul 2019 - 5:42

score 175 · Answer 12 · 2026-03-25 09:39:54

175

Views

Find Cauchy principal value of the following integral : $\int\limits_{\mathbb{R}}\frac{e^{-x^3}}{x+3}dx$

Published on 25 Mar 2026 - 9:39

score 88 · Answer 13 · 2026-03-25 09:40:00

88

Views

Prove that Cauchy principal value of $\int\limits_{\mathbb{R}}\frac{e^{-x^2}}{x+2}dx=\frac{π\text{erfi}{2}}{e^4}$

Published on 25 Mar 2026 - 9:40

score 603 · Answer 14 · 2026-03-25 01:18:48

603

Views

The closed-form solution of the family $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(pn+m)}$?

Published on 25 Mar 2026 - 1:18

score 150 · Answer 15 · 2026-03-25 22:03:23

150

Views

Find the integral in a closed form : $\int\limits_0^\infty\text{arctanh}(x^{2})e^{-ax^{2}}dx$

Published on 25 Mar 2026 - 22:03

List Question

evaluate the summation : $\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+1)(n+2x+3)}$

Generalizing Oksana's trilogarithm relation to $\text{Li}_3(\frac{n}8)$?

Is there a closed-form solution for $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(3n+m)}$?

Find triple summation rel in a closed form $S=\sum_{n=1}^{\infty}\sum_{m=1}^{n}\sum_{k=1}^{m}\frac{1}{(n+1)(k+1)(m+1)nmk}$

some identity of summation and generalization

Evaluate $\int_0^{\frac{\pi}{4}}\frac{1+\cos x-x^{2}}{(1+x\sin x)\sqrt {1-x^{2}}}\,dx$

Evaluate the integral : $\int_{0}^{\infty}e^{-2x}Ei^{2}(x)dx$

Expressing $\sum_{n = 1}^\infty \sum_{k = 1}^n \frac{1}{n^4 k\,2^k}$ as a finite sum involving $\zeta(\cdot)$, $Li_k(\cdot)$, $\pi$, and $\ln 2$

Prove ${_4 F_3} \left(\frac12, \frac12, 1, 1; \frac34, \frac54, \frac32; \frac14 \right)= \frac14 \left(\frac{\pi^2}{4}+\log^2 (2+\sqrt{3} ) \right)$

Closed form for $\sum _{k=0}^n (-1)^k \binom{n}{k} \binom{n+k}{n} (H_n-H_k) x^k$

Finding a closed form for recurrence relations $a_n=na_{n-1}+1$ and $a_n=na_{n-1}+n$

Find Cauchy principal value of the following integral : $\int\limits_{\mathbb{R}}\frac{e^{-x^3}}{x+3}dx$

Prove that Cauchy principal value of $\int\limits_{\mathbb{R}}\frac{e^{-x^2}}{x+2}dx=\frac{π\text{erfi}{2}}{e^4}$

The closed-form solution of the family $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(pn+m)}$?

Find the integral in a closed form : $\int\limits_0^\infty\text{arctanh}(x^{2})e^{-ax^{2}}dx$

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