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

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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 $\pi$

$\blacksquare$


Sources