Marco Cantarini

 46 Why does the hard-looking integral $\int_{0}^{\infty}\frac{x\sin^2(x)}{\cosh(x)+\cos(x)}dx=1$? 24 An integration of product $(1-x^n)$ 23 The sum of series with natural logarithm: $\sum_{n=1}^\infty \ln\left(\frac{n(n+2)}{(n+1)^2}\right)$ 21 Integral $\int_0^\infty\text{Li}_2\left(e^{-\pi x}\right)\arctan x\,dx$ 20 How to prove this approximation for a logarithm?

### Reputation (28,880)

 +10 How to show that $\sum_{n=1}^{\infty}\frac{H_n}{n^2+n}=\frac{\pi^2}{6}$ +10 Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$ +10 Concerning the sequence $\sum_{p \le n , p} \big\lfloor\frac{\log n}{\log p}\bigr\rfloor/n$ where p is prime +10 Limit of a sequence including infinite product. $\lim\limits_{n \to\infty}\prod_{k=1}^n \left(1+\frac{k}{n^2}\right)$

### Questions (10)

 18 Frullani 's theorem in a complex context. 12 On the integral $\int_{e}^{\infty}\frac{t^{1/2}}{\log^{1/2}\left(t\right)}\alpha^{-t/\log\left(t\right)}dt,\,\alpha>1.$ 9 About the product of two Elliptic integrals 6 The inverse Laplace transform of $e^{-z}\textrm{Ei}(z)z^{-1}\log(z).$ 4 About Mellin transform and harmonic series

### Tags (175)

 742 sequences-and-series × 307 229 summation × 95 730 integration × 214 216 number-theory × 81 715 calculus × 250 182 closed-form × 38 440 definite-integrals × 93 154 limits × 73 319 real-analysis × 132 146 analytic-number-theory × 87

### Accounts (4)

 Mathematics 28,880 rep 23373 Italian Language 194 rep 4 MathOverflow 179 rep 7 Stack Overflow 101 rep 1