Question

Given constrain $m=a_1>a_2>...>a_n$ and the elements are integer prove $\sum \frac{a_i-a_{i+1}}{a_i} < H_m$

score 50 · Answer 1 · 2024-03-16 10:03:20

50

Views

Given constrain $m=a_1>a_2>...>a_n$ and the elements are integer prove $\sum \frac{a_i-a_{i+1}}{a_i} < H_m$

Published on 16 Mar 2024 - 10:03

score 35 · Answer 2 · 2024-03-20 15:38:37

35

Views

What does the function $f(x) = \lim \limits_{n \to \infty} H_{\lfloor xn \rfloor} - H_n$ look like?

Published on 20 Mar 2024 - 15:38

score 80 · Answer 3 · 2015-01-03 21:49:45

Is there an easy way to find exact value of $\left\lfloor{H_{10^6}^\left({\frac12}\right)}\right\rfloor$ without using a calculator

score 1k · Answer 4 · 2015-01-08 15:21:52

1k

Views

Calculate $\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx$

Published on 08 Jan 2015 - 15:21

score 374 · Answer 5 · 2015-01-10 21:23:44

374

Views

How to evaluate $\sum _{n=1}^{\infty } \frac{(-1)^{n+1} H_{2 n}^{(2)}}{n} = 2\zeta(3) - \frac \pi 2 G- \frac {\pi }{48}\ln 2$?

Published on 10 Jan 2015 - 21:23

score 110 · Answer 6 · 2015-01-16 03:58:56

110

Views

Asymptotic of $\sum_{n = 1}^{N}\frac{1}{n}$ versus $\sum_{1 \leq n \leq x}\frac{1}{n}$

Published on 16 Jan 2015 - 3:58

score 225 · Answer 7 · 2015-01-19 04:12:34

225

Views

Question concerning the divergence of a kind of "hyperharmonic" series different than the definition of Conway and Guy

Published on 19 Jan 2015 - 4:12

score 371 · Answer 8 · 2015-01-19 07:51:56

371

Views

Does there exist a closed form for $L_k$ for any $k>3$?

Published on 19 Jan 2015 - 7:51

score 718 · Answer 9 · 2026-03-28 13:38:24

718

Views

Identity with Harmonic and Catalan numbers

Published on 28 Mar 2026 - 13:38

score 370 · Answer 10 · 2015-02-18 23:47:03

370

Views

Proof by induction that $\frac1{n+1} + \frac1{n+2} + \ldots + \frac1{2n} \geq \frac7{12}$

Published on 18 Feb 2015 - 23:47

score 122 · Answer 11 · 2015-02-20 11:37:30

122

Views

Is function $f(x)$ same as floor function?

Published on 20 Feb 2015 - 11:37

score 100 · Answer 12 · 2015-02-24 19:30:55

100

Views

Harmonic summation

Published on 24 Feb 2015 - 19:30

score 171 · Answer 13 · 2015-02-25 06:26:44

171

Views

Mathematical induction for inequalities: $\frac1{n+1} + \frac1{n+2} + \cdots +\frac1{3n+1} > 1$

Published on 25 Feb 2015 - 6:26

score 144 · Answer 14 · 2026-03-25 04:00:10

144

Views

Sum $\sum _{k=0}^{\infty } \frac{(-1)^k \psi (k+1)}{\left(k-\frac{3}{2}\right)^2}$

Published on 25 Mar 2026 - 4:00

score 178 · Answer 15 · 2015-03-01 21:29:09

178

Views

Confused by a proof about harmonic numbers

Published on 01 Mar 2015 - 21:29

List Question

Given constrain $m=a_1>a_2>...>a_n$ and the elements are integer prove $\sum \frac{a_i-a_{i+1}}{a_i} < H_m$

What does the function $f(x) = \lim \limits_{n \to \infty} H_{\lfloor xn \rfloor} - H_n$ look like?

Is there an easy way to find exact value of $\left\lfloor{H_{10^6}^\left({\frac12}\right)}\right\rfloor$ without using a calculator

Calculate $\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx$

How to evaluate $\sum _{n=1}^{\infty } \frac{(-1)^{n+1} H_{2 n}^{(2)}}{n} = 2\zeta(3) - \frac \pi 2 G- \frac {\pi }{48}\ln 2$?

Asymptotic of $\sum_{n = 1}^{N}\frac{1}{n}$ versus $\sum_{1 \leq n \leq x}\frac{1}{n}$

Question concerning the divergence of a kind of "hyperharmonic" series different than the definition of Conway and Guy

Does there exist a closed form for $L_k$ for any $k>3$?

Identity with Harmonic and Catalan numbers

Proof by induction that $\frac1{n+1} + \frac1{n+2} + \ldots + \frac1{2n} \geq \frac7{12}$

Is function $f(x)$ same as floor function?

Harmonic summation

Mathematical induction for inequalities: $\frac1{n+1} + \frac1{n+2} + \cdots +\frac1{3n+1} > 1$

Sum $\sum _{k=0}^{\infty } \frac{(-1)^k \psi (k+1)}{\left(k-\frac{3}{2}\right)^2}$

Confused by a proof about harmonic numbers

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