Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences of positive real numbers. Let $s_{n}=\sum_{i=1}^{n} a_{i}$.

(a) Show that if $\sum a_{n}$ and $\sum b_{n}$ converge then so does $\sum\left(a_{n}^{2}+b_{n}^{2}\right)^{1 / 2}$.

(b) Show that if $\sum a_{n}$ converges then $\sum \sqrt{a_{n} a_{n+1}}$ converges. Is the converse true?

(c) Show that if $\sum a_{n}$ diverges then $\sum \frac{a_{n}}{s_{n}}$ diverges. Is the converse true?

$[$ For part (c), it may help to show that for any $N \in \mathbb{N}$ there exist $m \geqslant n \geqslant N$ with

$$\left.\frac{a_{n+1}}{s_{n+1}}+\frac{a_{n+2}}{s_{n+2}}+\ldots+\frac{a_{m}}{s_{m}} \geqslant \frac{1}{2} .\right]$$