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:

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

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 $1 \ \text {(iv)}$