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

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Example of Convergent Complex Sequence

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

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

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

$\blacksquare$


Sources