Convergent Complex Series/Examples/1 over n^2 - i n

Example of Convergent Complex Series

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

$a_n = \dfrac 1 {n^2 - i n}$

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

$=$

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

$\ds $

$=$

$\ds \sum_{n \mathop = 1}^\infty \cmod {\dfrac {n^2 + i n} {n^4 + n^2} }$

$\ds $

$\le$

$\ds \sum_{n \mathop = 1}^\infty \cmod {\dfrac {n^2 + i n} {n^4} }$

$\ds $

$=$

$\ds \sum_{n \mathop = 1}^\infty \cmod {\dfrac {n + i} {n^3} }$

$\ds $

$\le$

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

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

$\blacksquare$

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