# Convergent Complex Sequence/Examples/(cos pi over n + i sin pi over n)^2n+1

## Example of Convergent Complex Sequence

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

$z_n = \paren {\cos \dfrac \pi n + i \sin \dfrac \pi n}^{2 n + 1}$

Then:

$\displaystyle \lim_{n \mathop \to \infty} z_n = 1$

## Proof

 $\ds z_n$ $=$ $\ds \paren {\cos \dfrac \pi n + i \sin \dfrac \pi n}^{2 n + 1}$ $\ds$ $=$ $\ds \cos \dfrac {\paren {2 n + 1} \pi} n + i \sin \dfrac {\paren {2 n + 1} \pi} n$ De Moivre's Theorem $\ds$ $=$ $\ds \cos \paren {2 + \dfrac 1 n} \pi + i \sin \paren {2 + \dfrac 1 n} \pi$ $\ds$ $\to$ $\ds \cos 2 \pi + i \sin 2 \pi$ as $\dfrac 1 n$ is a Basic Null Sequence $\ds$ $=$ $\ds 1$ Cosine of Multiple of Pi, Sine of Multiple of Pi

$\blacksquare$