Question

Prove that $ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \mathcal{O}(\log(n)) $.

score 1.6k · Answer 1 · 2013-02-24 22:48:31

1.6k

Views

Prove that $ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \mathcal{O}(\log(n)) $.

Published on 24 Feb 2013 - 22:48

score 1.6k · Answer 2 · 2013-02-26 03:31:42

1.6k

Views

A series involves harmonic number

Published on 26 Feb 2013 - 3:31

score 9.1k · Answer 3 · 2026-03-26 22:16:10

9.1k

Views

Showing that $\lim\limits_{n\to\infty}\sum^n_{k=1}\frac{1}{k}-\ln(n)=0.5772\ldots$

Published on 26 Mar 2026 - 22:16

score 398 · Answer 4 · 2026-03-28 05:22:36

398

Views

How can I evaluate $\int _0^1\frac{\text{Li}_2\left(-x\right)\ln \left(1-x\right)}{1+x}\:dx$

Published on 28 Mar 2026 - 5:22

score 82 · Answer 5 · 2020-08-04 10:15:21

82

Views

Back-and-forth harmonic progression -- do we get all of $\mathbb{Q}?$

Published on 04 Aug 2020 - 10:15

score 406 · Answer 6 · 2026-03-28 05:23:22

406

Views

How to evaluate $\int _0^1\frac{\ln ^2\left(1-x\right)\ln ^3\left(1+x\right)}{1+x}\:dx$

Published on 28 Mar 2026 - 5:23

score 140 · Answer 7 · 2020-08-06 14:10:21

140

Views

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

Published on 06 Aug 2020 - 14:10

score 929 · Answer 8 · 2026-03-28 05:23:26

929

Views

Evaluating $\int_0^1\frac{\arctan x\ln\left(\frac{2x^2}{1+x^2}\right)}{1-x}dx$

Published on 28 Mar 2026 - 5:23

score 280 · Answer 9 · 2026-03-28 05:22:43

280

Views

Is there a closed form without MZV for $ \sum _{k=1}^{\infty }\frac{H_k}{k^6\:2^k}$?

Published on 28 Mar 2026 - 5:22

score 89 · Answer 10 · 2026-03-29 21:36:35

89

Views

Maple bug? How to calculate the asymptotic expansion of the generalised harmonic function?

Published on 29 Mar 2026 - 21:36

score 152 · Answer 11 · 2020-08-14 09:03:23

152

Views

How to approach $\sum _{k\ge 1}\frac{\left(-1\right)^k\:H_k}{\left(2k+1\right)^2}$

Published on 14 Aug 2020 - 9:03

score 66 · Answer 12 · 2026-03-26 07:53:53

66

Views

Symbolic sum similar to Zeta function or harmonic number

Published on 26 Mar 2026 - 7:53

score 340 · Answer 13 · 2020-08-15 16:31:08

340

Views

Computing $\sum_{n=1}^\infty\frac{2^{2n}H_{n+1}}{(n+1)^2{2n\choose n}}$

Published on 15 Aug 2020 - 16:31

score 612 · Answer 14 · 2020-08-15 22:38:47

612

Views

Is there a closed form for $\sum_{n=1}^\infty\frac{2^{2n}H_n}{n^3{2n\choose n}}?$

Published on 15 Aug 2020 - 22:38

score 203 · Answer 15 · 2020-08-17 02:32:33

203

Views

Verifying the closed form of $\sum_{n=1}^\infty\frac{4^nH_n^{(2)}}{n^2{2n\choose n}}$

Published on 17 Aug 2020 - 2:32

List Question

Prove that $ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \mathcal{O}(\log(n)) $.

A series involves harmonic number

Showing that $\lim\limits_{n\to\infty}\sum^n_{k=1}\frac{1}{k}-\ln(n)=0.5772\ldots$

How can I evaluate $\int _0^1\frac{\text{Li}_2\left(-x\right)\ln \left(1-x\right)}{1+x}\:dx$

Back-and-forth harmonic progression -- do we get all of $\mathbb{Q}?$

How to evaluate $\int _0^1\frac{\ln ^2\left(1-x\right)\ln ^3\left(1+x\right)}{1+x}\:dx$

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

Evaluating $\int_0^1\frac{\arctan x\ln\left(\frac{2x^2}{1+x^2}\right)}{1-x}dx$

Is there a closed form without MZV for $ \sum _{k=1}^{\infty }\frac{H_k}{k^6\:2^k}$?

Maple bug? How to calculate the asymptotic expansion of the generalised harmonic function?

How to approach $\sum _{k\ge 1}\frac{\left(-1\right)^k\:H_k}{\left(2k+1\right)^2}$

Symbolic sum similar to Zeta function or harmonic number

Computing $\sum_{n=1}^\infty\frac{2^{2n}H_{n+1}}{(n+1)^2{2n\choose n}}$

Is there a closed form for $\sum_{n=1}^\infty\frac{2^{2n}H_n}{n^3{2n\choose n}}?$

Verifying the closed form of $\sum_{n=1}^\infty\frac{4^nH_n^{(2)}}{n^2{2n\choose n}}$

Trending Questions

Popular # Hahtags

Popular Questions