I want to prove that if $(a_n)$ is a complex sequence such that $\sum a_n = 0 $ then I can define an holomorphic function $f(z)$ on $ D:= \{z\ | \ |z|<1\} $ by $f(z)= \sum a_n z^n$
In order to do this I wanted to consider the subsequence $(a_{n_k})$ of the nonzero terms of $(a_n)$ and hence use a converse of the ratio test, i.e. say that $\lim_{k \rightarrow \infty}{|a_{n_{k+1}}/a_{n_k}}|\leq 1$
Hence it would be easy to prove that $f$ actually is holomorphic in $D$
My concern is that $\lim_{k \rightarrow \infty}{|a_{n_{k+1}}/a_{n_k}}|$ might not be defined (even if the terms are all non zero..) any suggestion on how to prove it rigorously?