# Terms in Convergent Series Converge to Zero

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## Contents

## 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:

\(\displaystyle a_N\) | \(=\) | \(\displaystyle \paren {a_1 + a_2 + \cdots + a_{N - 1} + a_N} - \paren {a_1 + a_2 + \cdots + a_{N - 1} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle s_N - s_{N - 1}\) | |||||||||||

\(\displaystyle \) | \(\to\) | \(\displaystyle s - s = 0 \text{ as } N \to \infty\) |

Hence the result.

$\blacksquare$

## Also see

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 4.3$. Series - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 6.9$ - 1992: Larry C. Andrews:
*Special Functions of Mathematics for Engineers*... (previous) ... (next): $\S 1.2.2$: Summary of convergence tests: Theorem $1.1$