Let $\left(a_{n}\right),\left(b_{n}\right)$ be sequences of real numbers. Let $S_{n}=\sum_{j=1}^{n} a_{j}$ and set $S_{0}=0$. Show that for any $1 \leqslant m \leqslant n$ we have

$$\sum_{j=m}^{n} a_{j} b_{j}=S_{n} b_{n}-S_{m-1} b_{m}+\sum_{j=m}^{n-1} S_{j}\left(b_{j}-b_{j+1}\right)$$

Suppose that the series $\sum_{n \geqslant 1} a_{n}$ converges and that $\left(b_{n}\right)$ is bounded and monotonic. Does $\sum_{n \geqslant 1} a_{n} b_{n}$ converge?

Assume again that $\sum_{n \geqslant 1} a_{n}$ converges. Does $\sum_{n \geqslant 1} n^{1 / n} a_{n}$ converge?

Justify your answers.

[You may use the fact that a sequence of real numbers converges if and only if it is a Cauchy sequence.]