Does the convergence of $\sum_{n = 0}^{\infty} |a_{n}| \Rightarrow |a_{n}| \rightarrow 0$ Consider the power series $\sum_{n = 0}^{\infty} a_{n} x^{n}$.
If $\sum_{n = 0}^{\infty} |a_{n}|, n \in \mathbb{Z}$ is absolutely convergent, does that necessarily imply $|a_{n}| \rightarrow 0$? 
When I think about it, it seems true. I'm not asking for a proof or anything I just want to get an intuition of why that is true (if it is to begin with).

Suppose $|a_n|$ converged to some nonzero number $L>0$. That implies that for any $\epsilon>0$, there is a natural number $N$ such that $L-\epsilon<|a_n|$ for all $n\ge N$. But then we would have
$$\sum_{n=0}^{\infty}|a_n|\ge \sum_{n=N}^{\infty}|a_n|>\sum_{n=N}^{\infty}(L-\epsilon)=\infty.$$
Contradiction.
