Divergent Series/Examples/(n + i) over n^2

Example of Divergent Series
The complex series defined as:
 * $\displaystyle S = \sum_{n \mathop = 1}^\infty \dfrac {n + i} {n^2}$

is divergent.

Proof
From Harmonic Series is Divergent, $\displaystyle \sum_{n \mathop = 1}^\infty \dfrac 1 n$ is a divergent series.

The result follows from Convergence of Series of Complex Numbers by Real and Imaginary Part.