20. Complex Sequence

Convergence of a Complex Sequence

Let $(z_n)$ be a complex sequence
Let $L\in \mathbb{C}$
Then $(z_n)$ converges to $L$ as $n\to \infty$ if

$$\forall ϵ>0:\exists N\in \mathbb{N}\text{ such that }\forall n\ge N:|z_n-L|<ϵ$$

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10 - Convergence of Complex Sequences

Convergence of Complex Series

Let $(z_n)$ be a complex sequence
Represent $z_n=x_n+i{y}_n$ with $x_n,y_n\in \mathbb{R}$ so $(x_n)$ and $(y_n)$ are real sequences
Then $(z_n)$ converges if and only iff

$$(x_n)\text{ and }(y_n)\text{ both converge}$$

Moreover if $(z_n)$ converges then

$$\underset{n\to \infty }{\lim }{z}_n=\underset{n\to \infty }{\lim }{x}_n+i\underset{n\to \infty }{\lim }{y}_n$$

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