Let $a_{n} \in \mathbb{R}$ for $n \geqslant 1$. What does it mean to say that the infinite series $\sum_{n} a_{n}$ converges to some value $A$ ? Let $s_{n}=a_{1}+\cdots+a_{n}$ for all $n \geqslant 1$. Show that if $\sum_{n} a_{n}$ converges to some value $A$, then the sequence whose $n$-th term is

$$\left(s_{1}+\cdots+s_{n}\right) / n$$

converges to some value $\tilde{A}$ as $n \rightarrow \infty$. Is it always true that $A=\tilde{A}$ ? Give an example where $\left(s_{1}+\cdots+s_{n}\right) / n$ converges but $\sum_{n} a_{n}$ does not.