Question

Prove $\sum_{k=1}^r\frac{1}{k+r}=\sum_{k=1}^{2r}\frac{(-1)^{k-1}}{k}$

score 159 · Answer 1 · 2020-02-17 20:53:04

159

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Prove $\sum_{k=1}^r\frac{1}{k+r}=\sum_{k=1}^{2r}\frac{(-1)^{k-1}}{k}$

Published on 17 Feb 2020 - 20:53

score 1.5k · Answer 2 · 2026-03-25 14:31:40

1.5k

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An approximation related to Euler's constant and the Harmonic number

Published on 25 Mar 2026 - 14:31

score 850 · Answer 3 · 2020-02-19 05:47:28

850

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On the alternating quadratic Euler sum $\sum_{n = 1}^\infty \frac{(-1)^n H_n H_{2n}}{n^2}$

Published on 19 Feb 2020 - 5:47

score 147 · Answer 4 · 2020-02-20 21:06:43

147

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How to prove $\sum _{k=1}^{\infty } (-1)^k H_{\frac{2 k}{3}} = -\frac{\pi }{2 \sqrt{3}}+\frac{3 \pi }{8}-\frac{3}{4} \log (2)$?

Published on 20 Feb 2020 - 21:06

score 196 · Answer 5 · 2020-02-23 20:22:23

196

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Evaluating $\int_0^1\frac{\ln x\operatorname{Li}_2(x)\ln(1-x/2)}{x}\ dx$

Published on 23 Feb 2020 - 20:22

score 1.1k · Answer 6 · 2020-02-28 17:44:21

1.1k

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Inequalities to give bounds on generalised harmonic numbers?

Published on 28 Feb 2020 - 17:44

score 32 · Answer 7 · 2026-03-25 12:50:17

32

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Is there any possibility that there exists a formula for the partial sums of the harmonic series? If not, why?

Published on 25 Mar 2026 - 12:50

score 186 · Answer 8 · 2020-02-29 14:02:59

How to prove the asymptotic expansion $\overline{H}_n \sim \log(2) -(-1)^n\left (\frac{1}{2n}-\frac{1}{4 n^2} +\frac{1}{8n^4}\mp\ldots\right)$?

score 61 · Answer 9 · 2026-03-25 11:32:57

61

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Question on proof of relationships between $f(s)=\frac{s}{s+1}$ and the analytic Harmonic number function $H(s)$

Published on 25 Mar 2026 - 11:32

score 709 · Answer 10 · 2026-03-25 14:34:36

709

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Closed form of Euler-type sum over zeta functions $\sum _{k=2}^{\infty } \frac{\zeta (k)}{k^2}$?

Published on 25 Mar 2026 - 14:34

score 69 · Answer 11 · 2020-03-06 10:54:32

69

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How to prove the result of summing the divergent asymptotic sum for $H_n$?

Published on 06 Mar 2020 - 10:54

score 320 · Answer 12 · 2020-03-07 06:40:34

320

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Evaluating $\int\limits_0^\infty x\operatorname{sech}^3x\ln(\operatorname{sech}x)\ dx$

Published on 07 Mar 2020 - 6:40

score 71 · Answer 13 · 2020-03-11 08:34:14

71

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Efficient method to approximate Euler sums to high precision

Published on 11 Mar 2020 - 8:34

score 244 · Answer 14 · 2020-03-13 03:41:40

244

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Sum involving Harmonic numbers and binomial coefficients

Published on 13 Mar 2020 - 3:41

score 278 · Answer 15 · 2026-03-26 13:02:21

278

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The closed form for $\sum_{n=1}^\infty \frac{H_{n/2}}{n^2}x^n$

Published on 26 Mar 2026 - 13:02

List Question

Prove $\sum_{k=1}^r\frac{1}{k+r}=\sum_{k=1}^{2r}\frac{(-1)^{k-1}}{k}$

An approximation related to Euler's constant and the Harmonic number

On the alternating quadratic Euler sum $\sum_{n = 1}^\infty \frac{(-1)^n H_n H_{2n}}{n^2}$

How to prove $\sum _{k=1}^{\infty } (-1)^k H_{\frac{2 k}{3}} = -\frac{\pi }{2 \sqrt{3}}+\frac{3 \pi }{8}-\frac{3}{4} \log (2)$?

Evaluating $\int_0^1\frac{\ln x\operatorname{Li}_2(x)\ln(1-x/2)}{x}\ dx$

Inequalities to give bounds on generalised harmonic numbers?

Is there any possibility that there exists a formula for the partial sums of the harmonic series? If not, why?

How to prove the asymptotic expansion $\overline{H}_n \sim \log(2) -(-1)^n\left (\frac{1}{2n}-\frac{1}{4 n^2} +\frac{1}{8n^4}\mp\ldots\right)$?

Question on proof of relationships between $f(s)=\frac{s}{s+1}$ and the analytic Harmonic number function $H(s)$

Closed form of Euler-type sum over zeta functions $\sum _{k=2}^{\infty } \frac{\zeta (k)}{k^2}$?

How to prove the result of summing the divergent asymptotic sum for $H_n$?

Evaluating $\int\limits_0^\infty x\operatorname{sech}^3x\ln(\operatorname{sech}x)\ dx$

Efficient method to approximate Euler sums to high precision

Sum involving Harmonic numbers and binomial coefficients

The closed form for $\sum_{n=1}^\infty \frac{H_{n/2}}{n^2}x^n$

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