User Redundant Aunt

44 votes

3 answers

2k views

Show that $e^{\sqrt 2}$ is irrational

sequences-and-series irrational-numbers

asked Sep 18, 2015 at 17:02

24 votes

1 answer

1k views

Proof of $\zeta(2)=\frac{\pi^2}{6}$

asked Apr 28, 2015 at 19:19

16 votes

2 answers

403 views

How to compute the following integral in $n$ variables?

calculus products

asked Nov 5, 2014 at 0:35

15 votes

2 answers

315 views

How to prove $\int_{0}^{-1} \frac{\operatorname{Li}_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $

calculus integration definite-integrals closed-form polylogarithm

asked Oct 31, 2014 at 19:53

15 votes

4 answers

801 views

Does it make sense to define $ \aleph_{\infty}=\lim\limits_{n\to\infty}\aleph_n $? Is its cardinality "infinitely infinite"?

set-theory cardinals infinity

asked Oct 9, 2015 at 18:57

14 votes

2 answers

1k views

How to compute the monstrous $ \int_0^{\frac{e-1}{e}}{\frac{x(2-x)}{(1-x)}\frac{\log\left(\log\left(1+\frac{x^2}{2-2x}\right)\right)}{2-2x+x^2}dx} $

calculus integration definite-integrals closed-form

asked Aug 21, 2015 at 22:33

13 votes

2 answers

1k views

Ways to prove Euler's formula for $\zeta(2n)$

sequences-and-series solution-verification riemann-zeta bernoulli-numbers

asked Jun 12, 2015 at 12:49

12 votes

3 answers

1k views

Function which takes every value uncountably often

real-analysis analysis elementary-set-theory

asked Dec 14, 2015 at 8:40

11 votes

1 answer

167 views

Conjecture about natural number satisfying $ m(n)^k+1\space\mid\space n^{2k}+1 $

number-theory prime-numbers

asked Dec 5, 2015 at 10:21

11 votes

4 answers

411 views

How to solve the functional equation $ f(f(x))=ax^2+bx+c $

functional-analysis functions functional-equations

asked Dec 31, 2015 at 14:23

11 votes

4 answers

433 views

Infinite sum involving number of solutions to $k=i^2+j^2$

sequences-and-series

asked Jun 7, 2015 at 18:28

10 votes

1 answer

439 views

How to prove that $\sum_{k=0}^\infty \binom{x}{x-k}\cdot\binom{x}{k-x} = 1$?

sequences-and-series binomial-coefficients

asked Jul 14, 2014 at 8:08

10 votes

3 answers

1k views

Harmonic number identity

calculus sequences-and-series binomial-coefficients harmonic-numbers beta-function

asked Oct 25, 2014 at 0:01

10 votes

2 answers

215 views

How to solve the functional equation $f\left(x^2+f(y)\right)=y+f(x)^2$

contest-math functional-equations

asked Sep 8, 2015 at 10:08

9 votes

2 answers

105 views

On $\mathcal{A}$ with $\forall C\in\mathcal{P}_{\infty}(\mathbb{N})\exists A\in\mathcal{A}:A\cap C\in\mathcal{P}_{\infty}(\mathbb{N})$

sequences-and-series convergence-divergence set-theory filters

asked Jul 23, 2019 at 6:44

8 votes

3 answers

1k views

Hard inequality $ (xy+yz+zx)\left(\frac{1}{(x+y)^2}+\frac{1}{(y+z)^2}+\frac{1}{(z+x)^2}\right)\ge\frac{9}{4} $

inequality contest-math substitution symmetric-polynomials uvw

asked Oct 5, 2014 at 19:36

8 votes

3 answers

374 views

How to compute the integral $ I\left(c\right)=\int_{0}^{1}{\frac{\ln(1-cx)}{1+x}dx} $

calculus integration definite-integrals

asked Jun 24, 2015 at 11:00

8 votes

2 answers

905 views

What is the regularity of the greatest eigenvalue of the Hessian matrix?

calculus real-analysis hessian-matrix

asked Jun 27, 2016 at 18:27

8 votes

2 answers

157 views

Elegant proof of combinatorial statement: at least $l-k+1$ children who were in larger teams

combinatorics discrete-mathematics

asked Oct 24, 2017 at 12:39

7 votes

5 answers

4k views

How to show that $(W^\bot)^\bot=W$ (in a finite dimensional vector space)

linear-algebra inner-products orthogonality

asked Mar 9, 2016 at 22:57

7 votes

1 answer

173 views

Integral involving hyperfactorial

calculus definite-integrals gamma-function zeta-functions

asked Oct 22, 2014 at 21:49

7 votes

3 answers

245 views

Calculate trigonometric integral $ \int_{-\infty}^{\infty}{\sin(x^2)}\,dx$

calculus trigonometry improper-integrals

asked Oct 6, 2014 at 23:03

7 votes

1 answer

135 views

How to solve $\frac{f^{-1}(x)f(x)}{x}=\frac{f^{-1}(x)+f(x)}{2}$?

functional-equations

asked Sep 10, 2015 at 17:40

6 votes

2 answers

1k views

Variation of the Josephus problem

number-theory

asked Aug 24, 2015 at 16:31

6 votes

1 answer

99 views

Differential equation related to Golomb's sequence

ordinary-differential-equations

asked Aug 7, 2015 at 15:20

6 votes

0 answers

207 views

generalization of geometric series $ \sum_{k=0}^\infty x^{p(k)} $

sequences-and-series polynomials closed-form

asked Oct 27, 2014 at 16:18

6 votes

2 answers

247 views

Inequality $\frac{\sqrt a+\sqrt b+\sqrt c}{2}\ge\frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}$ with weird condition

inequality contest-math

asked Sep 10, 2014 at 21:10

6 votes

1 answer

287 views

How to evaluate the following infinite sum involving the Riemann zeta function?

sequences-and-series closed-form zeta-functions

asked Sep 27, 2014 at 8:21

6 votes

3 answers

470 views

How to prove $ \sum_{r=1}^{k-1} \binom{k}{r}\cdot r^r \cdot (k-r)^{k-r-1} = k^k-k^{k-1} $

sequences-and-series

asked Oct 30, 2014 at 10:30

6 votes

4 answers

137 views

Is g with $\cosh(xg(x))=x\cosh(g(x))$ decreasing?

calculus real-analysis implicit-differentiation implicit-function-theorem

asked Jun 8, 2016 at 20:52

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187 Questions

Show that $e^{\sqrt 2}$ is irrational

Proof of $\zeta(2)=\frac{\pi^2}{6}$

How to compute the following integral in $n$ variables?

How to prove $\int_{0}^{-1} \frac{\operatorname{Li}_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $

Does it make sense to define $ \aleph_{\infty}=\lim\limits_{n\to\infty}\aleph_n $? Is its cardinality "infinitely infinite"?

How to compute the monstrous $ \int_0^{\frac{e-1}{e}}{\frac{x(2-x)}{(1-x)}\frac{\log\left(\log\left(1+\frac{x^2}{2-2x}\right)\right)}{2-2x+x^2}dx} $

Ways to prove Euler's formula for $\zeta(2n)$

Function which takes every value uncountably often

Conjecture about natural number satisfying $ m(n)^k+1\space\mid\space n^{2k}+1 $

How to solve the functional equation $ f(f(x))=ax^2+bx+c $

Infinite sum involving number of solutions to $k=i^2+j^2$

How to prove that $\sum_{k=0}^\infty \binom{x}{x-k}\cdot\binom{x}{k-x} = 1$?

Harmonic number identity

How to solve the functional equation $f\left(x^2+f(y)\right)=y+f(x)^2$

On $\mathcal{A}$ with $\forall C\in\mathcal{P}_{\infty}(\mathbb{N})\exists A\in\mathcal{A}:A\cap C\in\mathcal{P}_{\infty}(\mathbb{N})$

Hard inequality $ (xy+yz+zx)\left(\frac{1}{(x+y)^2}+\frac{1}{(y+z)^2}+\frac{1}{(z+x)^2}\right)\ge\frac{9}{4} $

How to compute the integral $ I\left(c\right)=\int_{0}^{1}{\frac{\ln(1-cx)}{1+x}dx} $

What is the regularity of the greatest eigenvalue of the Hessian matrix?

Elegant proof of combinatorial statement: at least $l-k+1$ children who were in larger teams

How to show that $(W^\bot)^\bot=W$ (in a finite dimensional vector space)

Integral involving hyperfactorial

Calculate trigonometric integral $ \int_{-\infty}^{\infty}{\sin(x^2)}\,dx$

How to solve $\frac{f^{-1}(x)f(x)}{x}=\frac{f^{-1}(x)+f(x)}{2}$?

Variation of the Josephus problem

Differential equation related to Golomb's sequence

generalization of geometric series $ \sum_{k=0}^\infty x^{p(k)} $

Inequality $\frac{\sqrt a+\sqrt b+\sqrt c}{2}\ge\frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}$ with weird condition

How to evaluate the following infinite sum involving the Riemann zeta function?

How to prove $ \sum_{r=1}^{k-1} \binom{k}{r}\cdot r^r \cdot (k-r)^{k-r-1} = k^k-k^{k-1} $

Is g with $\cosh(xg(x))=x\cosh(g(x))$ decreasing?