# Sequences of Complex Numbers

A sequence of real numbers is an infinite ordered list $(a_n)_{n=1}^{\infty}$ of real numbers. We can similarly define a sequence of complex numbers.

Definition: A Sequence of Complex Numbers is an infinite order list of complex numbers $(z_n)_{n=1}^{\infty} = (z_1, z_2, ..., z_n, ...)$ where $z_n \in \mathbb{C}$ for each $n \in \mathbb{N}$. Each $z_n$ is called a Term of the sequence. |

*We can denote any sequence of complex numbers with a different starting index if convenient.*

For example, the first few terms of the complex sequence $(n + i^n)_{n=1}^{\infty}$ are:

(1)Many of the definitions familiar with sequences of real numbers are readily extended to sequences of complex numbers.

Definition: A sequence of complex numbers $(z_n)_{n=1}^{\infty}$ is said to Converge to $Z \in \mathbb{C}$ if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then $\mid z_n - Z \mid < \epsilon$. If the sequence $(z_n)_{n=1}^{\infty}$ does not converge to any $Z \in \mathbb{C}$ then $(z_n)_{n=1}^{\infty}$ is said to Diverge. |

For example, consider the sequence $\left ( \frac{i^n}{n} \right)_{n=1}^{\infty}$. The first few terms of this sequence are:

(2)We claim that this sequence converges to $Z = 0$. To prove this, let $\epsilon > 0$ be given. Then we want $\biggr \lvert \frac{i^n}{n} - 0 \biggr \rvert < \epsilon$. Now:

(3)For any given $\epsilon > 0$, choose $N \in \mathbb{N}$ such that $\displaystyle{N > \frac{1}{\epsilon}}$. Then if $n \geq N$ we have that:

(4)In other words, for $n \geq N$:

(5)Therefore the sequence $\left ( \frac{i^n}{n} \right )_{n=1}^{\infty}$ converges to $Z = 0$.

We will now state some properties of limits of sequences of complex numbers. All of these statements are proved analogously to the proofs for the real cases.

Theorem 1: If $(z_n)_{n=1}^{\infty}$ is a sequence of complex numbers that converges then its limit is unique. |

Theorem 2: Let $(z_n)_{n=1}^{\infty}$ and $(w_n)_{n=1}^{\infty}$ be sequences of real numbers that converge to $Z$ and $W$ respectively. Then:a) $(z_n + w_n)_{n=1}^{\infty}$ converges to $Z + W$.b) $(z_n - w_n)_{n=1}^{\infty}$ converges to $Z - W$.c) $(z_nw_n)_{n=1}^{\infty}$ converges to $Z \cdot W$.d) $\displaystyle{\left ( \frac{z_n}{w_n} \right )_{n=1}^{\infty}}$ converges to $\displaystyle{\frac{Z}{W}}$ provided that $W \neq 0$. |