# Convergent Complex Sequence/Examples/((1 + i n) over (1 + n))^3

## Example of Convergent Complex Sequence

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

$z_n = \paren {\dfrac {1 + i n} {1 + n} }^3$

Then:

$\displaystyle \lim_{n \mathop \to \infty} z_n = -i$

## Proof

 $\ds z_n$ $=$ $\ds \paren {\dfrac {1 + i n} {1 + n} }^3$ $\ds$ $=$ $\ds \paren {\dfrac {\frac 1 n + i} {\frac 1 n + 1} }^3$ $\ds$ $\to$ $\ds \paren {\dfrac i 1}^3$ as $\dfrac 1 n$ is a Basic Null Sequence $\ds$ $=$ $\ds i^3$ $\ds$ $=$ $\ds -i$

$\blacksquare$