TITLE: Calculating the infinite series $1-\frac13+\frac15-\frac17+\frac19-\frac1{11}\cdots$
QUESTION [5 upvotes]: My main question is the one in the title, however I was also wondering, in general when it comes to infinite series, how can you find out whether the series converges to a value or not? And can you tell if there will be something strange about it? What I mean is that you would expect the sum of all natural numbers to be infinity, but it is -1/12. Is there a way of knowing if something like this will happen?
Sorry for the ton of questions! And thank you for any answers :)

REPLY [3 votes]: An other way:
$$\arctan x=\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}x^n.$$
if $|x|<1$. Morevoer $$\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$$
is an Alternating series, then it converge. By Abel's theorem you can conclude that
$$\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}=\arctan(1)=\frac{\pi}{4}.$$