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

Example of Convergent Complex Sequence

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

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

Then:

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

Proof

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

$\blacksquare$