11 A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent 9 Show a detail prove of : $\int_{0}^{1}\int_{0}^{1}\left({x\over 1-xy}\cdot{\ln{x}-\ln{y}\over \ln{x}+\ln{y}}\right)\mathrm dx\mathrm dy=1-2\gamma$ 6 Find the value of $\int\limits_0^{+\infty} \frac{(\coth x-1)(x\coth x-1)}{x} dx$ 6 About the integral $\int_0^{\pi/2} \cos^n(x)\cos(nx) \;dx$ and $\int_0^{\pi/2} \cos^n(x)\sin(nx) \;dx$. 5 Double series convergent to $2\zeta(4)$?

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### Questions (15)

 19 Mathematical Explanation of Mathematica Summation ${\sum_{n=1}^{\infty}\frac{(2n-1)!}{(2n+2)!}\zeta(2n)}$ 9 Expressing Zeta function using Gamma series 8 $\lim\limits_{s\to0^+}\sum_n\frac{\cos\left(\pi\frac{n}{m}\right)}{n^s}$ & $\lim\limits_{s\to0^+}\sum_n\frac{\sin\left(\pi\frac{n}{m}\right)}{n^s}$ 6 On the sum: $\sum\limits_{n=0}^{\infty}\left[\,\sum\limits_{k=1}^{a}\frac{1}{an+k}-\sum\limits_{k=1}^{b}\frac{1}{bn+k}\,\right]$ 4 An Integral Representation of Logarithmic Derivative of Zeta Function

### Tags (83)

 86 sequences-and-series × 67 22 limits × 10 39 calculus × 18 15 trigonometry × 6 35 integration × 12 13 summation × 11 23 definite-integrals × 6 12 inequality × 7 22 real-analysis × 18 11 equidistribution