# Divergent Complex Sequence/Examples/(2 over 3 + 3i over 4)^n

## Example of Divergent Complex Sequence

Let $\sequence {z_n}$ be the complex sequence defined as:

$z_n = \paren {\dfrac 2 3 + \dfrac {3 i} 4}^n$

Then $\ds \lim_{n \mathop \to \infty} z_n$ does not exist.

## Proof

 $\ds \cmod {z_n}^2$ $=$ $\ds \cmod {\dfrac 2 3 + \dfrac {3 i} 4}^{2 n}$ $\ds$ $=$ $\ds \paren {\dfrac 4 9 + \dfrac 9 {16} }^n$ Definition of Complex Modulus $\ds$ $=$ $\ds \paren {\dfrac {145} {144} }^n$ $\ds$ $\to$ $\ds \infty$ as $\cmod {\dfrac 2 3 + \dfrac {3 i} 4} > 1$

Thus $\cmod {z_n} \to \infty$ and so the limit does not exist.

$\blacksquare$