Series of functions of a complex variable.

Definition 7.2.1   Let be given a sequence $(f_n(z))_{n \geq 0}$ of functions of a complex variable $z$ . Suppose that all the functions $f_n$ are defined on the same domain $D$ . The expression

$$\underset{k=0}{\overset{+\infty }{\sum}} f_k(z) =f_0(z)+f_1(z)+f_2(z)+ \dots + f_n(z) + \dots$$

is called the infinite series with $f_n(z)$ as its general term.

In fact for every $z \in D$ , we definehere a series of complex numbers.

Definition 7.2.2   The sequence $(S_n)_{n \geq 0}$ whose general term is $S_n=\underset{k=0}{\overset{n}{\sum}} f_k = f_0+f_1 + \dots + f_n$ is called the sequence of partial sums of the given series.