State and prove the Comparison Test for real series.

Assume $0 \leqslant x_{n}<1$ for all $n \in \mathbb{N}$. Show that if $\sum x_{n}$ converges, then so do $\sum x_{n}^{2}$ and $\sum \frac{x_{n}}{1-x_{n}}$. In each case, does the converse hold? Justify your answers.

Let $\left(x_{n}\right)$ be a decreasing sequence of positive reals. Show that if $\sum x_{n}$ converges, then $n x_{n} \rightarrow 0$ as $n \rightarrow \infty$. Does the converse hold? If $\sum x_{n}$ converges, must it be the case that $(n \log n) x_{n} \rightarrow 0$ as $n \rightarrow \infty$ ? Justify your answers.