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.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $2 \ \text {(iii)}$