Convergence of functions is a very broad subject because it is important to specify in what sense the sequence converges.

Here are two examples:

1) (version of the Arzela Ascoli theorem) suppose a series of continuous functions  \sum f_n(x)  on a compact metric space  K (e.g. a closed interval) converges pointwise and moreover the partial sums are equicontinuous (i.e. for every \epsilon > 0 and x_0 \in K, there is  delta independent of  N  such that for d(x, x_0) < delta, we have  |\sum_{n \le N} f_n(x_0) - \sum_{n \le N} f_n(x)| < \epsilon.) , then the sum function is continuous.

2)  A sequence of holomorphic functions that is uniformly convergent on compacta is holomorphic

The  ratio test for a series is actually an application of example 2. It still works if  the f_n are holomorohic and there is some constant  such that f_n(x) \le const |x|^n to  It does not work for Dirichlet series \sum a_n n^{-s} but example 2 works fine to prove that the sum defines a holomorphic function.  

@George Stoica. Yes to prove pointwise convergence that certainly works.