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:
- $\ds \lim_{n \mathop \to \infty} z_n = -1$
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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $1 \ \text {(iv)}$