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CHAMSI
  • Member for 4 years, 1 month
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5 votes
0 answers
57 views

Proof verification : A weird identity involving sums

5 votes
2 answers
205 views

A closed form for $ \int_{0}^{\frac{\pi}{2}}{\left(\frac{\sin{\left(nx\right)}}{\sin{x}}\right)^{2p}\,\mathrm{d}x} $?

4 votes
1 answer
229 views

Proving $\prod\limits_{k=1}^{n-1}\left(1-\frac{\sin^2(x/2n)}{\sin^2(k\pi/2n)}\right)=\frac{\sin{x}}{n\sin(x/n)}$ and related tangent formula

3 votes
3 answers
114 views

Find $ \lim\limits_{x\to 0}{\frac{1-\sqrt{1+x}\sqrt[3]{1-x}\cdots\sqrt[2n+1]{1-x}}{x}} $ without using L'Hopital's rule.

3 votes
1 answer
142 views

Combinatorial proof of $ \sum\limits_{k=0}^n(-1)^k\frac{2^{n-k}\binom{n}{k}}{(m+k+1)\binom{m+k}{m}}=\sum\limits_{k=0}^n\frac{\binom{n}{k}}{m+k+1}$

2 votes
4 answers
290 views

Prove that : $ \int_{0}^{2\ln{\varphi}}{\theta\ln{\left(2\sinh{\frac{\theta}{2}}\right)}\,\mathrm{d}\theta}=-\frac{1}{5}\zeta\left(3\right) $

2 votes
3 answers
128 views

Proving that $ \prod\limits_{k=1}^{n}{\left(1+\frac{1}{k^{3}}\right)}<\mathrm{e} $

1 vote
3 answers
105 views

Prove $ \int\limits_{0}^{+\infty}{\frac{\mathrm{d}x}{\cosh^{n}{x}}}=\int\limits_{0}^{\frac{\pi}{2}}{\cos^{n-1}{x}\,\mathrm{d}x} $

1 vote
1 answer
68 views

Derivatives of Gamma at 1

1 vote
1 answer
134 views

Other series representations for zeta and eta functions

0 votes
1 answer
123 views

Math for fun : Find the following limits without using L'Hôpital's rule or series expansion :