Convergent Complex Sequence/Examples/(1 over 2 + i 4 over 5)^n

Example of Convergent Complex Sequence

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

$z_n = \paren {\dfrac 1 2 + i \dfrac 4 5}^n$

Then:

$\ds \lim_{n \mathop \to \infty} z_n = 0$

$\ds \cmod {z_n}^2$

$=$

$\ds \cmod {\dfrac 1 2 + i \dfrac 4 5}^{2 n}$

$\ds $

$=$

$\ds \paren {\dfrac 1 4 + \dfrac {16} {25} }^n$

$\ds $

$=$

$\ds \paren {\dfrac {25 + 64} {100} }^n$

$\ds $

$=$

$\ds \paren {\dfrac {89} {100} }^n$

$\ds $

$\to$

$\ds 0$

as $\cmod {\dfrac 2 3 + \dfrac {3 i} 4} < 1$

Thus $\cmod {z_n} \to 0$ and so $z_n \to 0$ as $n \to \infty$.

$\blacksquare$

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $1 \ \text {(iii)}$