# Divergent Series/Examples/n over n^2 + i

## Example of Divergent Series

The complex series defined as:

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

is divergent.

## Proof

 $\ds S$ $=$ $\ds \sum_{n \mathop = 1}^\infty \frac n {n^2 + i}$ $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty \frac {n \paren {n^2 - i} } {\paren {n^2 + i} \paren {n^2 - i} }$ $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty \frac {n^3 - i n^2} {n^4 + 1}$ $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty \paren {\frac {n^3} {n^4 + 1} - \frac {i n^2} {n^4 + 1} }$

Then we have:

 $\ds \sum_{n \mathop = 1}^\infty \frac {n^3} {n^4 + 1}$ $\ge$ $\ds \sum_{n \mathop = 1}^\infty \frac {n^3} {2 n^4}$ $\ds$ $=$ $\ds \frac 1 2 \sum_{n \mathop = 1}^\infty \frac 1 n$

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

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

$\blacksquare$