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:
- $\ds \lim_{n \mathop \to \infty} z_n = 1$
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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.2$. Sequences: Example $\text {(v)}$