Question

Proving that $(-1)^{n+1}G_n =\int_0^\infty \frac{1}{(1+x)^n (\pi^2+\ln^2 x)} dx$

score 467 · Answer 1 · 2018-10-05 14:32:23

467

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Proving that $(-1)^{n+1}G_n =\int_0^\infty \frac{1}{(1+x)^n (\pi^2+\ln^2 x)} dx$

Published on 05 Oct 2018 - 14:32

score 3.9k · Answer 2 · 2026-03-25 06:30:19

3.9k

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We have $5$ red, $3$ blue and $2$ green balls. We take $3$ of them at random. What is the probability that they are of different colors?

Published on 25 Mar 2026 - 6:30

score 96 · Answer 3 · 2018-10-11 00:23:02

96

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$a + b = 2$ implies $a^c + b^c \ge 2$ for any real $c \ge 1$

Published on 11 Oct 2018 - 0:23

score 1.2k · Answer 4 · 2018-10-13 07:59:37

1.2k

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If $\sin\alpha+\sin\beta+\sin\gamma=\cos\alpha+\cos\beta+\cos\gamma=0$, then find the values of $\sum \sin^2 \alpha$ and $\sum \cos^2 \alpha$.

Published on 13 Oct 2018 - 7:59

score 353 · Answer 5 · 2018-10-15 10:20:45

353

Views

Is this seemingly new proof that $[0, 1]$ is compact correct?

Published on 15 Oct 2018 - 10:20

score 105 · Answer 6 · 2026-03-25 17:26:14

105

Views

Reformulating the definition of compactness

Published on 25 Mar 2026 - 17:26

score 46 · Answer 7 · 2018-10-18 07:15:04

46

Views

Maps of posets with $f(x)<g(x)$ induce homotopic functions

Published on 18 Oct 2018 - 7:15

score 502 · Answer 8 · 2026-03-25 22:26:54

502

Views

Show that $\int\limits_0^{\frac{\pi}2}4\cos^2(x)\log^2(\cos x)~\mathrm dx=-\pi\log 2+\pi\log^2 2-\frac{\pi}2+\frac{\pi^3}{12}$

Published on 25 Mar 2026 - 22:26

score 495 · Answer 9 · 2026-03-25 09:28:14

495

Views

The cyclic quadrilateral and the slopes of its sides

Published on 25 Mar 2026 - 9:28

score 86 · Answer 10 · 2018-10-21 16:21:15

86

Views

Prove $\frac{2^{4n}-(-1)^n}{17} \in \mathbb{N}$ by induction

Published on 21 Oct 2018 - 16:21

score 82 · Answer 11 · 2018-10-22 13:33:45

82

Views

Generalisation of $A^{k}$ matrix

Published on 22 Oct 2018 - 13:33

score 267 · Answer 12 · 2026-03-25 12:45:50

267

Views

Modelling a linear minimization program

Published on 25 Mar 2026 - 12:45

score 34 · Answer 13 · 2018-10-23 02:26:05

34

Views

Let A be a set, totally ordered under relation ≤. Let x ∈ A. Then x is a maximal element of A if and only if x is the greatest element of A.

Published on 23 Oct 2018 - 2:26

score 927 · Answer 14 · 2026-03-25 15:49:11

927

Views

Comparing logarithms with different bases

Published on 25 Mar 2026 - 15:49

score 171 · Answer 15 · 2026-03-25 19:02:26

171

Views

Irreducible factors in an arithmetic sequence

Published on 25 Mar 2026 - 19:02

List Question

Proving that $(-1)^{n+1}G_n =\int_0^\infty \frac{1}{(1+x)^n (\pi^2+\ln^2 x)} dx$

We have $5$ red, $3$ blue and $2$ green balls. We take $3$ of them at random. What is the probability that they are of different colors?

$a + b = 2$ implies $a^c + b^c \ge 2$ for any real $c \ge 1$

If $\sin\alpha+\sin\beta+\sin\gamma=\cos\alpha+\cos\beta+\cos\gamma=0$, then find the values of $\sum \sin^2 \alpha$ and $\sum \cos^2 \alpha$.

Is this seemingly new proof that $[0, 1]$ is compact correct?

Reformulating the definition of compactness

Maps of posets with $f(x)<g(x)$ induce homotopic functions

Show that $\int\limits_0^{\frac{\pi}2}4\cos^2(x)\log^2(\cos x)~\mathrm dx=-\pi\log 2+\pi\log^2 2-\frac{\pi}2+\frac{\pi^3}{12}$

The cyclic quadrilateral and the slopes of its sides

Prove $\frac{2^{4n}-(-1)^n}{17} \in \mathbb{N}$ by induction

Generalisation of $A^{k}$ matrix

Modelling a linear minimization program

Let A be a set, totally ordered under relation ≤. Let x ∈ A. Then x is a maximal element of A if and only if x is the greatest element of A.

Comparing logarithms with different bases

Irreducible factors in an arithmetic sequence

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