Series of complex numbers.

Definition 7.1.1   Let be given a sequence $(u_n)_{n \geq 0}$ of complex numbers. The expression

$$\underset{k=0}{\overset{+\infty }{\sum}} u_k =u_0+u_1+u_2+ \dots + U_n + \dots$$

is called the infinite series with $u_n$ as its general term.

Definition 7.1.2   The sequence $(S_n)_{n \geq 0}$ whose general term is $S_n=\underset{k=0}{\overset{n}{\sum}} u_k = u_0+u_1 + \dots + u_n$ is called the sequence of partial sums of the given series.