If a series converges, does the sequence of its summands necessarily converge to zero?

Question

A sequence is defined as $a_{1}, a_{2}, ... $, and $\sum\limits_{k=1}^\infty a_n$ converges. Is it necessarily true that $ \lim\limits_{n \rightarrow \infty } a_{n} = 0$?

I don't know where to begin. It seems intuitively true but I've come across enough non-intuitive results to stop taking that at face value. I can also show that the converse is definitely not true.

Answer 1

If the partial sums converge to $L$, they differ from $L$ by less than your favourite $\epsilon>0$, provided you go far enough in. But each term is a difference of two partial sums, so will be $<2\epsilon$. So yes, they do go to $0$.

Answer 2

If $A_n=\sum_{k=1}^na_k$, then the limit $\lim_{n\to\infty}A_n$ exists and therefore $\lim_{n\to\infty}A_n-A_{n-1}=0$. But them$$0=\lim_{n\to\infty}(A_n-A_{n-1})=\lim_{n\to\infty}a_n.$$