$\sum_{n=-\infty}^{\infty} \ z_n$ what is mean of this series converge

Question

let $\sum_{n=-\infty}^{\infty} \ z_n$ where $z_n \in \mathbb{C}$ .

what does it mean to say this series converge?? what is sequence of its pratial sums?

is it $S_N=\sum_{n=-N}^{n=N} \ z_n$ where $N\in \mathbb{N}$

Answer 1

Good question .Easiest way is to say $\sum_{n=-\infty}^{\infty}z_n$= $\sum_{n=-\infty}^{m}z_n$ + $\sum_{n=m+1}^{\infty}z_n$ and use the usual partial sums definition separately for each of the sums on the right .For convergence of the sum from $-\infty$ to $\infty$ it is required that both sums on the right converge. Easily the answer for the sum on the left is independent of of m . Your guess is much to weak to be of use .