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 , with sum equal to if

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

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

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