Question

Compute $2\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^4}+\sum_{n=1}^\infty\frac{H_nH_n^{(3)}}{n^3}$

score 281 · Answer 1 · 2019-08-10 06:22:05

281

Views

Compute $2\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^4}+\sum_{n=1}^\infty\frac{H_nH_n^{(3)}}{n^3}$

Published on 10 Aug 2019 - 6:22

score 1.1k · Answer 2 · 2019-08-16 23:21:43

1.1k

Views

How to evaluate $\int_0^1\frac{\ln x\ln^2(1-x)}{1+x}dx$ in an elegant way?

Published on 16 Aug 2019 - 23:21

score 502 · Answer 3 · 2019-08-19 19:25:57

502

Views

Mind-blowing Sums: Compute $\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^22^n}$ and $\sum_{n=1}^\infty\frac{H_n^3}{n^22^n}$

Published on 19 Aug 2019 - 19:25

score 283 · Answer 4 · 2026-03-25 09:37:13

283

Views

Divergent sums via analytic continuation: power series vs Dirichlet series

Published on 25 Mar 2026 - 9:37

score 189 · Answer 5 · 2019-08-25 04:10:21

189

Views

Closed form of $\int_{0}^1 \operatorname{Li}^2_n(x) dx$ for positive integer $n$

Published on 25 Aug 2019 - 4:10

score 902 · Answer 6 · 2019-08-28 08:08:32

902

Views

How to compute $\sum_{n=1}^\infty\frac{H_n^2}{n^32^n}$?

Published on 28 Aug 2019 - 8:08

score 169 · Answer 7 · 2019-09-06 06:15:58

169

Views

Prove $\int_0^1\frac{\ln x\operatorname{Li}_2(-x)}{x(1-x^2)}dx=\frac{17}{16}\zeta(4)$

Published on 06 Sep 2019 - 6:15

score 756 · Answer 8 · 2019-09-06 23:47:31

756

Views

Challenging sum: Compute $\sum_{n=1}^\infty\frac{H_{2n}H_n^{(2)}}{(2n)^2}$

Published on 06 Sep 2019 - 23:47

score 460 · Answer 9 · 2019-09-12 03:14:08

460

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Very advanced sums: Compute $\sum _{n=1}^{\infty } \frac{H_n H_{2 n}^{(2)}}{(2 n)^2}$ and $\sum _{n=1}^{\infty } \frac{H_n H_{2 n}^2}{(2 n)^2}$

Published on 12 Sep 2019 - 3:14

score 444 · Answer 10 · 2019-09-17 23:26:49

444

Views

Compute $\sum_{n=1}^\infty\frac{H_nH_{2n}}{(2n+1)^3}$

Published on 17 Sep 2019 - 23:26

score 1.6k · Answer 11 · 2019-09-22 20:31:22

1.6k

Views

A group of important generating functions involving harmonic number.

Published on 22 Sep 2019 - 20:31

score 276 · Answer 12 · 2019-09-25 18:00:16

276

Views

Is there a closed form for the integral $ \int_{0}^{1}\frac{\mathrm{Li}_2(-x^2)}{x^2+1} \mathrm{d}x $

Published on 25 Sep 2019 - 18:00

score 432 · Answer 13 · 2019-09-27 18:32:28

432

Views

Advanced Sum: Compute $\sum_{n=1}^\infty\frac{H_{2n}H_n^{(2)}}{(2n+1)^2}$

Published on 27 Sep 2019 - 18:32

score 283 · Answer 14 · 2019-09-29 04:41:14

283

Views

How to compute $\int_0^1\left(\operatorname{Li}_2(x)-\zeta(2)\right)\frac{\ln^2(1-x^2)}{1-x^2}\ dx$

Published on 29 Sep 2019 - 4:41

score 22 · Answer 15 · 2014-03-19 01:20:38

22

Views

interpretation of logarithmic in the dimension

Published on 19 Mar 2014 - 1:20

List Question

Compute $2\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^4}+\sum_{n=1}^\infty\frac{H_nH_n^{(3)}}{n^3}$

How to evaluate $\int_0^1\frac{\ln x\ln^2(1-x)}{1+x}dx$ in an elegant way?

Mind-blowing Sums: Compute $\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^22^n}$ and $\sum_{n=1}^\infty\frac{H_n^3}{n^22^n}$

Divergent sums via analytic continuation: power series vs Dirichlet series

Closed form of $\int_{0}^1 \operatorname{Li}^2_n(x) dx$ for positive integer $n$

How to compute $\sum_{n=1}^\infty\frac{H_n^2}{n^32^n}$?

Prove $\int_0^1\frac{\ln x\operatorname{Li}_2(-x)}{x(1-x^2)}dx=\frac{17}{16}\zeta(4)$

Challenging sum: Compute $\sum_{n=1}^\infty\frac{H_{2n}H_n^{(2)}}{(2n)^2}$

Very advanced sums: Compute $\sum _{n=1}^{\infty } \frac{H_n H_{2 n}^{(2)}}{(2 n)^2}$ and $\sum _{n=1}^{\infty } \frac{H_n H_{2 n}^2}{(2 n)^2}$

Compute $\sum_{n=1}^\infty\frac{H_nH_{2n}}{(2n+1)^3}$

A group of important generating functions involving harmonic number.

Is there a closed form for the integral $ \int_{0}^{1}\frac{\mathrm{Li}_2(-x^2)}{x^2+1} \mathrm{d}x $

Advanced Sum: Compute $\sum_{n=1}^\infty\frac{H_{2n}H_n^{(2)}}{(2n+1)^2}$

How to compute $\int_0^1\left(\operatorname{Li}_2(x)-\zeta(2)\right)\frac{\ln^2(1-x^2)}{1-x^2}\ dx$

interpretation of logarithmic in the dimension

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