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

Then $\displaystyle \lim_{n\to\infty} a_n = 0$.

Proof

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

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

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

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle a_N\)	\(=\)	\(\displaystyle \left({a_1 + a_2 + \cdots + a_{N-1} + a_N}\right) - \left({a_1 + a_2 + \cdots + a_{N-1} }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle s_N - s_{N-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\to\)	\(\displaystyle s - s = 0 \text{ as } N \to \infty\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Hence the result.

$\blacksquare$

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 6.9$