What is the definition of $\sum_{n = - \infty}^{\infty} a_n$

Question

I know that in Complex Analysis we use sums of the form $\displaystyle\sum_{n = - \infty}^{\infty} a_n$

What is the actual meaning of this symbol? I expect that under some nice enough conditions, it would be equal to something like $\displaystyle \sum_{n=1}^\infty b_n$, where $b_{2k} = a_k$ and $b_{2k+1} = a_{-k}$. But what is the actual definition?

Answer 1

The sum is defined as $$\displaystyle\sum_{n = - \infty}^{\infty} a_n:= \lim_{N\to\infty}\sum_{n=0}^{N} a_n+\lim_{M\to\infty}\sum_{n=1}^{M} a_{-n}$$