Terms in Convergent Series Converge to Zero

Theorem
Let $\left \langle {a_n} \right \rangle$ be a sequence in any of the standard number fields $\Q$, $\R$, or $\C$.

Suppose that the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges in any of the standard number fields $\Q$, $\R$, or $\C$.

Then:
 * $\displaystyle \lim_{n \mathop \to \infty} a_n = 0$

Proof
Let $\displaystyle s = \sum_{n \mathop = 1}^\infty a_n$.

Then $\displaystyle s_N = \sum_{n \mathop = 1}^N a_n \to s$ as $N \to \infty$.

Also, $s_{N - 1} \to s$ as $N \to \infty$.

Thus:

Hence the result.

Also see

 * Terms in Uniformly Convergent Series Converge Uniformly to Zero