Calculating the infinite series $1-\frac13+\frac15-\frac17+\frac19-\frac1{11}\cdots$ 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 :)
Recall that
$$
\frac{1}{1+x^2}=1-x^2+x^4+\cdots
$$
If you integrate from $0$ to $1$
$$
\begin{align}\int_0^1\frac{1}{1+x^2} dx&amp; =\int_0^11-x^2+x^4-x^6+\cdots\,dx\\
&amp; =1-\frac13+\frac15-\frac17+\cdots
\end{align}
$$ but $$\int_0^1\frac{1}{1+x^2} dx=\arctan 1-\arctan 0 = \frac\pi4, $$
then
$$
1-\frac13+\frac15-\frac17+\frac19-\cdots=\frac\pi4
$$
$$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+ \dots =\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$$

Have you got taught the Dirichlet's criterion? 

Use this,and you prove that your infinite sum converges.