Some problems (not all of them challenging, you may try)

Computation 1. Use simple trick taught in Math190 (something call convolution?) to show that $\displaystyle \left(\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n+1}}{2^{2n+1}(2n+1)!}\right)^{2}=-\frac{1}{2}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n)!}x^{2n}.$ I don't know if there is any trigonometric function that proves this formula instantly.

Problem 2. Determine if the series $\displaystyle \sum_{n=2}^\infty \frac{\log n}{n(n-1)}$ (base $e$) converges.

Problem 3. Find all $p\in\mathbb{R}$ such that $\displaystyle \sum_{k=2}^\infty \frac{1}{\big(\log (\log k)\big)^{p\log k}}$ converges.

Problem 4. Let $0<x_1<1$ and Define $x_{n+1}=x_n(1-x_n)$. Show that the series $\sum x_n$ diverges.