# Convergent Complex Series/Examples/e^in over n^2

## Example of Convergent Complex Series

The series $\ds \sum_{n \mathop = 1}^\infty a_n$, where:

$a_n = \dfrac {e^{i n} } {n^2}$

is convergent.

## Proof

 $\ds \sum_{n \mathop = 1}^\infty \cmod {\dfrac {e^{i n} } {n^2} }$ $=$ $\ds \sum_{n \mathop = 1}^\infty \cmod {\dfrac 1 {n^2} }$

Thus $\ds \sum_{n \mathop = 1}^\infty \dfrac {e^{i n} } {n^2}$ is absolutely convergent.

The result follows from Absolutely Convergent Series is Convergent: Complex Numbers.

$\blacksquare$