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

Example of Divergent Series

The complex series defined as:

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

is divergent.

Proof

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

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.

$\blacksquare$