Convergence and divergence.

Definition 7.2.3 (pointwise convergence)   The series $\underset{k=0}{\overset{+\infty }{\sum}} f_k(z)$ of functions of the complex variable $z$ is (pointwise) convergent if, for every $z \in D$ , the sequence of partial sums $S_n= \underset{k=0}{\overset{n}{\sum}} f_k(z)$ is convergent. The limit is called the sum of the series. A non convergent series is called divergent.

The series $\underset{k=0}{\overset{+\infty }{\sum}} f_k(z)$ is convergent of $D$ , with sum equal to $F(z)$ if

$$\forall z \in D, \forall \varepsilon>0 \exists N_0 \in \mathbb{N} s.t. n>N_0 \Longrightarrow \vert F(z)-S)n(z)\vert< \varepsilon.$$

Note that the natural number $N_0$ depends on $z$ and on $\varepsilon$ .