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Sangchul Lee's user avatar
Sangchul Lee's user avatar
Sangchul Lee's user avatar
Sangchul Lee
  • Member for 12 years, 4 months
  • Last seen more than 3 years ago
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I am an ex-mathematician who still enjoys solving math problems. Here is a selection of my answers:

Limiting distribution of $\frac{1}{n} \sum_{k=1}^{n}|S_{k-1}|(X_k^2 - 1)$, where $X_k$ are i.i.d. standard normal

Does the sum $\sum_{n \geq 1} \frac{2^n\text{ mod } n}{n^2}$ converge?

Expected absolute difference between two i.i.d. variables

Does the sequence $x_{n+1} = -16 + 6x_n + \frac{12}{x_n}$ converge?

Proving $\lim_{x \to 0+} \sum_{n=0}^\infty (-1)^n/(n!)^x = \frac{1}{2}$

Solution of the functional equation $f(x) + f(y) = f(x + y + 2f(xy))$

How to prove the existence of this limit?

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212
A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$
196
Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$
193
Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$
163
Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$
121
Prove that $\frac{100!}{50!\cdot2^{50}} \in \Bbb{Z}$

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