# Convergent Complex Sequence/Examples/(3+in)^2 over n^2

## Example of Convergent Complex Sequence

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

$z_n = \dfrac {\paren {3 + i n}^2} {n^2}$

Then:

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

## Proof

 $\ds z_n$ $=$ $\ds \dfrac {\paren {3 + i n}^2} {n^2}$ $\ds$ $=$ $\ds \dfrac {9 + 6 i n - n^2} {n^2}$ $\ds$ $=$ $\ds \dfrac {9 - n^2} {n^2} + \dfrac {6 i} n$ $\ds$ $=$ $\ds \dfrac 9 {n^2} - 1 + i \dfrac 6 n$ $\ds$ $\to$ $\ds 0 - 1 + i \times 0$ as both $\sequence {\dfrac 9 {n^2} }$ and $\sequence {\dfrac 6 n}$ are Basic Null Sequences

$\blacksquare$