# Divergent Series/Examples/((2+3i) over (3-2i))^n

## Example of Divergent Series

The complex series defined as:

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

is divergent.

## Proof

 $\ds \cmod {\paren {\dfrac {2 + 3 i} {3 - 2 i} }^n}$ $=$ $\ds \cmod {\dfrac {2 + 3 i} {3 - 2 i} }^n$ $\ds$ $=$ $\ds \paren {\dfrac {4 + 9} {9 + 4} }^{n/2}$ $\ds$ $=$ $\ds 1$

Thus the sequence $\sequence {\paren {\dfrac {2 + 3 i} {3 - 2 i} }^n}$ does not tend to zero.

Hence from Terms in Convergent Series Converge to Zero it follows that $S$ is divergent.

$\blacksquare$