A theorem says as follows
Let $A$ be an open subset of $\mathbb{C}$, and let $(f_n)_{n\geq 1}$ be a sequence of holomorphic functions on $A$. Assume that the infinite series $$s(z)=\sum_{i=1}^{\infty}f_i(z)$$ converges locally uniformly on $A$ to a function $s:A\to\mathbb{C}$. Then $s$ is holomorphic on $A$.
I made a list, I'd like to clarify.
- From beginning, if one writes "$s(z)=\sum_{i=1}^{\infty}f_i(z)$", is it understood that the series is (pointwise) convergent to some limit function $s:A\to\mathbb{C}$?
- The conclusion, where it says that $s$ is holomorphic on $A$; is it the limit function or the series that is holomorphic? [For example, if we are dealing with a geometric series.]
- Similar to the last point; if one has two functions $g,h$ that agree on an open set $B\subseteq \mathbb{C}$, and it is known that $g$ is holomorphic on $B$, does it then imply that $h$ is holomorphic on $B$ as well, or should one check it?
1) In this case, they tell you that the series converges locally uniformly at every point of $A,$ so in particular in converges at every point of $A.$ Typically, I think that if the series converges, the author says so, but if he just said that a function was given by a series on some domain, I'd take it to mean it converges at every point of that domain.
2) The function is holomorphic, as is the sum of the series. I don't know what a "holomorphic series" is.
3) Yes. They're the same function on $B;$ how could one be holomorphic and the other not?