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Atmos
  • Member for 4 years, 7 months
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798 votes
49 answers
125k views

Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem)

97 votes
3 answers
3k views

How prove this nice limit $\lim\limits_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$

50 votes
0 answers
904 views

probability for a $n\times n$ matrix to have no complex eigenvalues

  • 9,576
35 votes
4 answers
2k views

Relation between the roots of a function

  • 23.6k
32 votes
1 answer
735 views

Assume $f\left({\frac{f(x)}{x}}\right)= f(x)$. Show that $ f$ continuous

  • 3,574
28 votes
3 answers
8k views

Why does the boundary of the Mandelbrot set contain a cardioid?

  • 9,560
27 votes
7 answers
2k views

Convergence of $a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}}$

13 votes
5 answers
1k views

Solve $\int \limits_{0}^{\infty} \frac{\cos(x)}{\cosh(x)} dx$ without complex integration.

  • 1,045
9 votes
5 answers
494 views

Why are these determinants $0$?

  • 151
9 votes
3 answers
1k views

Integral $ \int_0^\infty \frac{\ln x}{(x+c)(x-1)} dx$

8 votes
6 answers
250 views

Calculate the closed form of the following series

  • 71
8 votes
2 answers
371 views

An integral inequality from AMM

  • 123
8 votes
1 answer
141 views

Evaluate integral $\int_{0}^{\frac\pi2}\frac{x}{\sin(x)+6x} \, dx$

  • 1,256
8 votes
2 answers
255 views

On the integral $\int_0^\infty{\frac{\sin(\lambda x)\mathrm{d}x}{e^{2\pi x}-1}}$

7 votes
0 answers
89 views

Polynomials such that $\frac{P(x^2)}{P(x)}$ is also a polynomial [duplicate]

  • 4,361
7 votes
1 answer
167 views

If $\sum\limits_{i=1}^na_i=\prod\limits_{i=1}^na_i$ for every $n$, identify $\lim\limits_{n\to \infty}a_n$

  • 711
7 votes
4 answers
213 views

Let $a$ be the real root of the equation $x^3+x+1=0$

  • 1,211
7 votes
2 answers
187 views

Probability question is bugging me

7 votes
1 answer
120 views

How to prove the closed form of the integral $\int \frac {dx}{\prod_{r=0}^n (x+r)}$

  • 9,412
7 votes
2 answers
185 views

Integral inequality $\frac{1}{2M}\le\int_0^1xf(x)dx\le1-\frac{1}{2M}$

7 votes
1 answer
82 views

Find a limit of $\ln(u_n)/2^n$

  • 6,955
6 votes
2 answers
273 views

What is the value of $f(100)$?

  • 22.3k
6 votes
4 answers
161 views

$(1+x+x^2)^n=P_0+P_{1}x + P_{2}x^2+ \cdots +P_{2n}x^{2n}$ Prove that,$ P_0+P_{3}+P_{6}+ \cdots =3^{(n-1)}$ [duplicate]

6 votes
1 answer
198 views

Change the trace of a Matrix

  • 969
5 votes
4 answers
113 views

How to find $f(m)=\prod\limits_{n=2}^{\infty}\left(1-\frac{1}{n^m}\right)^{-1}$ (if $m>1$)?

  • 1,465
5 votes
1 answer
104 views

Limit of Recursive Sequence $y_{n+1}=y_n+\frac{1}{ny_n}$ with $y_1=1$

5 votes
2 answers
136 views

Integrating $\int^\infty_ 0\frac{x^2}{e^{x^2} - 1}\,\mathrm dx$

5 votes
1 answer
107 views

Matrix eigenvalues

4 votes
2 answers
554 views

Closed form of definite integral from 2006 MIT Integration Bee [closed]

4 votes
1 answer
103 views

Does the series $\sum_{n=2}^{\infty}\frac{1}{1+(-1)^{n}\sqrt{n}}$ converge or diverge? [duplicate]