Question

Showing that $\frac{1}{18} + \frac{1}{19} + \cdots + \frac{1}{47} < 1$ without brute force calculation

score 446 · Answer 1 · 2026-03-25 04:41:47

446

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Showing that $\frac{1}{18} + \frac{1}{19} + \cdots + \frac{1}{47} < 1$ without brute force calculation

Published on 25 Mar 2026 - 4:41

score 241 · Answer 2 · 2020-01-29 17:57:56

241

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What's the generating function for $\sum_{n=1}^\infty\frac{\overline{H}_n}{n^3}x^n\ ?$

Published on 29 Jan 2020 - 17:57

score 158 · Answer 3 · 2020-01-30 02:16:49

158

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Computing $\sum_{n=1}^\infty(-1)^n\frac{\overline{H}_nH_n}{n^2}$

Published on 30 Jan 2020 - 2:16

score 436 · Answer 4 · 2020-01-30 22:15:26

436

Views

Evaluating $\int_0^1 \frac{\ln (1 - x) \operatorname{Li}_2 (-x)}{1 + x} \, dx$

Published on 30 Jan 2020 - 22:15

score 411 · Answer 5 · 2026-03-26 08:00:51

411

Views

Is there a closed form function for the integral $\int_{0}^{z} \frac{1}{x}\log(1-x)\log(x)\log(1+x)\,dx$?

Published on 26 Mar 2026 - 8:00

score 145 · Answer 6 · 2020-02-05 18:08:17

145

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Computing $\sum_{n=1}^\infty\frac{(-1)^{n-1}H_n^{(2)}}{(2n+1)^4}$

Published on 05 Feb 2020 - 18:08

score 95 · Answer 7 · 2026-03-25 14:35:54

95

Views

How to show that the integral $\int_0^1 \ln(1-x^{1/n})dx=-\sum_{k=1}^n\frac{1}{k}$

Published on 25 Mar 2026 - 14:35

score 239 · Answer 8 · 2020-02-10 02:16:31

239

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Is there closed form for $\sum_{k=1}^\infty (-1)^k\frac{H_k^{(4)}}{k^2}$?

Published on 10 Feb 2020 - 2:16

score 352 · Answer 9 · 2020-02-11 22:14:30

352

Views

A double sum for the square of the natural logarithm of $2$.

Published on 11 Feb 2020 - 22:14

score 191 · Answer 10 · 2020-02-12 16:05:46

191

Views

Prove $H_{\frac{1}{8}}=\int_{0}^{1}\frac{x^{\frac{1}{8}}-1}{x-1}dx$

Published on 12 Feb 2020 - 16:05

score 105 · Answer 11 · 2020-02-13 11:36:20

105

Views

How to compute $H_{5+\sqrt{7}}$

Published on 13 Feb 2020 - 11:36

score 142 · Answer 12 · 2020-02-13 17:12:54

142

Views

How to calculate the generating function of the unified harmonic sum $U(\sigma,n,p) = \sum_{k=1}^n \frac{\sigma^k}{k^p}$?

Published on 13 Feb 2020 - 17:12

score 376 · Answer 13 · 2020-02-13 19:57:16

376

Views

How to define the $r$th harmonic number if $r$ is any real number?

Published on 13 Feb 2020 - 19:57

score 112 · Answer 14 · 2026-03-26 10:58:33

112

Views

About Harmonic Numbers and a sequence in OEIS

Published on 26 Mar 2026 - 10:58

score 305 · Answer 15 · 2026-03-25 09:29:00

305

Views

Is the closed form for $\sum_{k=1}^\infty\frac{\overline{H}_k}{k^m}$ known in the literature?

Published on 25 Mar 2026 - 9:29

List Question

Showing that $\frac{1}{18} + \frac{1}{19} + \cdots + \frac{1}{47} < 1$ without brute force calculation

What's the generating function for $\sum_{n=1}^\infty\frac{\overline{H}_n}{n^3}x^n\ ?$

Computing $\sum_{n=1}^\infty(-1)^n\frac{\overline{H}_nH_n}{n^2}$

Evaluating $\int_0^1 \frac{\ln (1 - x) \operatorname{Li}_2 (-x)}{1 + x} \, dx$

Is there a closed form function for the integral $\int_{0}^{z} \frac{1}{x}\log(1-x)\log(x)\log(1+x)\,dx$?

Computing $\sum_{n=1}^\infty\frac{(-1)^{n-1}H_n^{(2)}}{(2n+1)^4}$

How to show that the integral $\int_0^1 \ln(1-x^{1/n})dx=-\sum_{k=1}^n\frac{1}{k}$

Is there closed form for $\sum_{k=1}^\infty (-1)^k\frac{H_k^{(4)}}{k^2}$?

A double sum for the square of the natural logarithm of $2$.

Prove $H_{\frac{1}{8}}=\int_{0}^{1}\frac{x^{\frac{1}{8}}-1}{x-1}dx$

How to compute $H_{5+\sqrt{7}}$

How to calculate the generating function of the unified harmonic sum $U(\sigma,n,p) = \sum_{k=1}^n \frac{\sigma^k}{k^p}$?

How to define the $r$th harmonic number if $r$ is any real number?

About Harmonic Numbers and a sequence in OEIS

Is the closed form for $\sum_{k=1}^\infty\frac{\overline{H}_k}{k^m}$ known in the literature?

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