Question

Computing $\sum_{n=1}^\infty\frac{(-1)^nH_n^2}{2n+1}$ in an alternative way

score 376 · Answer 1 · 2026-04-01 13:31:16

376

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Computing $\sum_{n=1}^\infty\frac{(-1)^nH_n^2}{2n+1}$ in an alternative way

Published on 01 Apr 2026 - 13:31

score 278 · Answer 2 · 2020-08-23 18:57:10

278

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Calculating $\int_0^1\frac{\ln^2x\ln(1-x)}{1-x}dx$ without using Beta function and Euler sum.

Published on 23 Aug 2020 - 18:57

score 97 · Answer 3 · 2026-04-09 18:16:04

97

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Identity for $\gamma$ and proving the identity in different ways .

Published on 09 Apr 2026 - 18:16

score 95 · Answer 4 · 2020-08-24 08:22:19

95

Views

Is this substitution, a sufficient condition to find the range of the function?

Published on 24 Aug 2020 - 8:22

score 151 · Answer 5 · 2026-03-25 23:42:37

151

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Given four real numbers $a,b,c,d$ so that $1\leq a\leq b\leq c\leq d\leq 3$. Prove that $a^2+b^2+c^2+d^2\leq ab+ac+ad+bc+bd+cd.$

Published on 25 Mar 2026 - 23:42

score 187 · Answer 6 · 2020-08-25 01:36:41

187

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Proving the existence of a bounded linear operator $A$ such that $Q(x) = \langle Ax, x\rangle$

Published on 25 Aug 2020 - 1:36

score 226 · Answer 7 · 2026-04-12 01:45:37

226

Views

Proof validation: Proving that the group of homeomorphisms of $[0,1]$ is infinite

Published on 12 Apr 2026 - 1:45

score 113 · Answer 8 · 2020-08-26 05:23:36

113

Views

Combinatoric proof of $\binom{j-1}{m-1}=\sum_{k=m}^j(-1)^{k-m}\binom{j}{k}$?

Published on 26 Aug 2020 - 5:23

score 334 · Answer 9 · 2026-04-13 11:21:26

334

Views

How to approach $\sum_{n=0}^\infty(-1)^n\frac{H_{2n+1}}{(2n+1)^3}$ elegantly?

Published on 13 Apr 2026 - 11:21

score 112 · Answer 10 · 2026-03-29 14:02:56

112

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Prove using induction: Given a directed graph like the one in the picture and a function $F(x,y)$ that returns the number of simple directed paths...

Published on 29 Mar 2026 - 14:02

score 332 · Answer 11 · 2026-04-01 15:01:22

332

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Alternative ways to evaluate $\int_0^1 x^{2n-1}\ln(1+x)dx =\frac{H_{2n}-H_n}{2n}$

Published on 01 Apr 2026 - 15:01

score 158 · Answer 12 · 2026-04-13 12:41:26

158

Views

Sine of sum proof

Published on 13 Apr 2026 - 12:41

score 76 · Answer 13 · 2020-08-30 20:02:27

76

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Any idea to prove that $\coth (x)-(1-y) \coth [x (1-y)]<\frac{4 x}{1-x^4}$?

Published on 30 Aug 2020 - 20:02

score 161 · Answer 14 · 2026-03-26 16:07:09

161

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Proving $\sum \frac{b+c}{9(a^2+3bc)+4(a+b+c)^2}\geqslant \frac{1}{4(a+b+c)}$

Published on 26 Mar 2026 - 16:07

score 197 · Answer 15 · 2026-03-25 17:44:34

197

Views

Proving ${\frac {35{x}^{2}+7x(y+z)+23yz}{35(x^2+y^2+z^2)+37(xy+yz+zx)}}\leqslant \sqrt {{\frac {{x}^{2}+yz}{6\,{y}^{2} +6\,yz+6\,{z}^{2}}}}$

Published on 25 Mar 2026 - 17:44

List Question

Computing $\sum_{n=1}^\infty\frac{(-1)^nH_n^2}{2n+1}$ in an alternative way

Calculating $\int_0^1\frac{\ln^2x\ln(1-x)}{1-x}dx$ without using Beta function and Euler sum.

Identity for $\gamma$ and proving the identity in different ways .

Is this substitution, a sufficient condition to find the range of the function?

Given four real numbers $a,b,c,d$ so that $1\leq a\leq b\leq c\leq d\leq 3$. Prove that $a^2+b^2+c^2+d^2\leq ab+ac+ad+bc+bd+cd.$

Proving the existence of a bounded linear operator $A$ such that $Q(x) = \langle Ax, x\rangle$

Proof validation: Proving that the group of homeomorphisms of $[0,1]$ is infinite

Combinatoric proof of $\binom{j-1}{m-1}=\sum_{k=m}^j(-1)^{k-m}\binom{j}{k}$?

How to approach $\sum_{n=0}^\infty(-1)^n\frac{H_{2n+1}}{(2n+1)^3}$ elegantly?

Prove using induction: Given a directed graph like the one in the picture and a function $F(x,y)$ that returns the number of simple directed paths...

Alternative ways to evaluate $\int_0^1 x^{2n-1}\ln(1+x)dx =\frac{H_{2n}-H_n}{2n}$

Sine of sum proof

Any idea to prove that $\coth (x)-(1-y) \coth [x (1-y)]<\frac{4 x}{1-x^4}$?

Proving $\sum \frac{b+c}{9(a^2+3bc)+4(a+b+c)^2}\geqslant \frac{1}{4(a+b+c)}$

Proving ${\frac {35{x}^{2}+7x(y+z)+23yz}{35(x^2+y^2+z^2)+37(xy+yz+zx)}}\leqslant \sqrt {{\frac {{x}^{2}+yz}{6\,{y}^{2} +6\,yz+6\,{z}^{2}}}}$

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