# Exchange limit and sum in certain conditions

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.

• Dosnt Cauchys formula with contour a small circle around 0 show that the series converges uniformly in a nbh of 0? – NJK Jun 2 '14 at 15:37
• Yes, this is correct. Thank you, NJK. – ebg Jun 3 '14 at 18:17