Question

Can one show that $\sum_{n=1}^N\frac{1}{n} -\log N - \gamma \leqslant \frac{1}{2N}$ without using the Euler-Maclaurin formula?

score 435 · Answer 1 · 2011-06-09 11:28:56

435

Views

Can one show that $\sum_{n=1}^N\frac{1}{n} -\log N - \gamma \leqslant \frac{1}{2N}$ without using the Euler-Maclaurin formula?

Published on 09 Jun 2011 - 11:28

score 19.3k · Answer 2 · 2011-07-20 05:03:34

19.3k

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Do harmonic numbers have a “closed-form” expression?

Published on 20 Jul 2011 - 5:03

score 10.6k · Answer 3 · 2011-10-18 04:03:43

10.6k

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The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$

Published on 18 Oct 2011 - 4:03

score 412 · Answer 4 · 2012-02-04 13:39:16

412

Views

Help needed with definite integral for $\lim_{n\rightarrow \infty}\sum_{k=1}^{n} \frac{1}{n+k}$

Published on 04 Feb 2012 - 13:39

score 396 · Answer 5 · 2012-02-29 23:16:58

396

Views

Show the sequence $a_{n} = \frac{1}{n+1} + \frac{1}{n+2} ... +\frac{1}{2n}$ converges to ln 2

Published on 29 Feb 2012 - 23:16

score 8.1k · Answer 6 · 2012-03-03 17:00:47

8.1k

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Proof by induction of summation inequality: $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac1{2^n}\ge 1+\frac{n}2$

Published on 03 Mar 2012 - 17:00

score 6.9k · Answer 7 · 2012-06-10 02:52:39

6.9k

Views

Showing inequality for harmonic series.

Published on 10 Jun 2012 - 2:52

score 3.1k · Answer 8 · 2012-07-01 19:13:50

3.1k

Views

Name Drawing Puzzle

Published on 01 Jul 2012 - 19:13

score 4.1k · Answer 9 · 2012-10-05 10:45:05

4.1k

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prove $\frac{1}{ n+1}+\frac{1}{ n+2}+\cdots+\frac{1}{2n}<\frac{25}{36}$ by mathematical induction

Published on 05 Oct 2012 - 10:45

score 539 · Answer 10 · 2012-10-25 20:53:42

539

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Find $\lim_{n\to\infty} (1+\frac{1}{2}+...+\frac{1}{n})\frac{1}{n}$

Published on 25 Oct 2012 - 20:53

score 124 · Answer 11 · 2012-12-05 15:32:37

124

Views

Explanation of the inequality $\sum_{k=1}^{n} 1/k \le 1 + \int_1^n(1/x) \, \mathrm{d} x$

Published on 05 Dec 2012 - 15:32

score 11.1k · Answer 12 · 2026-03-25 07:46:06

11.1k

Views

Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$

Published on 25 Mar 2026 - 7:46

score 3.5k · Answer 13 · 2013-01-30 02:06:58

3.5k

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Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$

Published on 30 Jan 2013 - 2:06

score 2.4k · Answer 14 · 2013-02-16 12:31:15

2.4k

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A series involving the harmonic numbers : $\sum_{n=1}^{\infty}\frac{H_n}{n^3}$

Published on 16 Feb 2013 - 12:31

score 52.2k · Answer 15 · 2013-02-17 18:01:30

52.2k

Views

Simple proof of showing the Harmonic number $H_n = \Theta (\log n)$

Published on 17 Feb 2013 - 18:01

List Question

Can one show that $\sum_{n=1}^N\frac{1}{n} -\log N - \gamma \leqslant \frac{1}{2N}$ without using the Euler-Maclaurin formula?

Do harmonic numbers have a “closed-form” expression?

The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$

Help needed with definite integral for $\lim_{n\rightarrow \infty}\sum_{k=1}^{n} \frac{1}{n+k}$

Show the sequence $a_{n} = \frac{1}{n+1} + \frac{1}{n+2} ... +\frac{1}{2n}$ converges to ln 2

Proof by induction of summation inequality: $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac1{2^n}\ge 1+\frac{n}2$

Showing inequality for harmonic series.

Name Drawing Puzzle

prove $\frac{1}{ n+1}+\frac{1}{ n+2}+\cdots+\frac{1}{2n}<\frac{25}{36}$ by mathematical induction

Find $\lim_{n\to\infty} (1+\frac{1}{2}+...+\frac{1}{n})\frac{1}{n}$

Explanation of the inequality $\sum_{k=1}^{n} 1/k \le 1 + \int_1^n(1/x) \, \mathrm{d} x$

Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$

Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$

A series involving the harmonic numbers : $\sum_{n=1}^{\infty}\frac{H_n}{n^3}$

Simple proof of showing the Harmonic number $H_n = \Theta (\log n)$

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