Let $\sum_{i=1}^{\infty} f_i(s)$ be a series of analytic functions and suppose it converges on some neighbourhood $V$ of $s=0$ and it converges uniformly on $V\cap\{s \;|\; |s|> \varepsilon \}$ for every $\varepsilon>0$. So it is analytic on the punctured neighbourhood $V-\{0\}$. Suppose $\lim_{s\to 0} \sum_{i=1}^{\infty} f_i(s)$ exists (and finite).
I am looking for references for conditions (other than uniform convergence in whole $V$) for which $ \lim_{s\to 0} \sum_{i=1}^{\infty} f_i(s) = \sum_{i=1}^{\infty} f_i(0) $
Thank you.
