Is $\sum_{n=0}^{\infty}2^n$ equal to $-1$? Why? 
Possible Duplicate:
Divisibility with sums going to infinity 

From Wikipedia and Minute Physics i see that the sum would be -1. I find this challenging to understand, how does a sum of positive integers end up being negative?

It's all about the principle of analytic continuation.  The function $f(x)=\sum_{n=0}^\infty z^n$ defines an analytic function in the unit disk, equal to the meromorphic function $g(z)=1/(1-z)$.  Note that the equality $f\equiv g$ holds only in the disk, where $f$ converges absolutely.  Despite this, if we naively wanted to assign any value to $f(x)$ outside of $\vert z \vert < 1$, an intuitive choice would be to set $f(2)=g(2)= -1$.  Moreover, the theory of complex functions implies that this is somehow the only possible choice (for this function, at least).
The principle is called analytic continuation, and it is incredibly important in many areas of mathematics, most notably complex analysis.
