TITLE: Can series start with $-∞$?
QUESTION [5 upvotes]: So, I decided to dig a little deeper into numerical integration because we hardly had any of that in my analysis class. I've come across this method for improper integrals:
Метод Самокиша (not in English, unfortunately).
What scares me though, is that the series in the last formula starts with $- \infty$. Is that even possible? We haven't studied series yet, but from what I understand the series usually starts with $1$ (or $0$) and goes to infinity.
I really hope you can help me figure this out. Thanks!

REPLY [3 votes]: What scares me though, is that the series in the last formula starts with $−\infty$. Is that even possible? We haven't studied series yet, but from what I understand the series usually starts with $1$ (or $0$) and goes to infinity.

You should not be scared.
Series of the type:
$$\sum_{n=-\infty}^\infty a_n \qquad \text{(also denoted by } \textstyle\sum_{n\in \mathbb{Z}} a_n\text{)}$$
are usually called bilateral series. A bilateral series converges iff the limit:
$$\lim_{N,M\to \infty} \sum_{n=-M}^N a_n$$
exists; otherwise, it is said to diverge. If you want, you can think a convergent bilateral series as a sum of two "standard" series, i.e.:
$$\sum_{n=-\infty}^\infty a_n = \sum_{n=0}^\infty a_n +\sum_{n=1}^\infty a_{-n}$$
For example, the bilateral series:
$$\sum_{n=-\infty}^\infty \frac{1}{(2n+1)^2}$$
converges: in fact, for fixed $N,M\in \mathbb{N}$ you get:
$$\begin{split} \lim_{N,M\to \infty} \sum_{n=-M}^N \frac{1}{(2n+1)^2} &= \lim_{N\to \infty} \sum_{n=0}^N \frac{1}{(2n+1)^2} +\lim_{M\to \infty} \sum_{n=1}^M \frac{1}{(1-2n)^2} \\ &= \sum_{n=0}^\infty \frac{1}{(2n+1)^2} +\sum_{n=1}^\infty \frac{1}{(2n-1)^2}\end{split}$$
for both series $\sum 1/(2n+1)^2$ and $\sum 1/(1-2n)^2$ converge; in particular:
$$\sum_{n=-\infty}^\infty \frac{1}{(2n+1)^2} =\frac{\pi^2}{4}\; .$$