Divergent series and $p$-adics If we naïvely apply the formula $$\sum_0^\infty a^i = {1\over 1-a}$$ when $a=2$, we get the silly-seeming claim that $1+2+4+\ldots = -1$. But  in the 2-adic integers, this formula is correct. 
Surely this is not a coincidence? What is the connection here?  Do the $p$-adics provide generally useful methods for summing divergent series?  I know very little about either divergent series or $p$-adic analysis; what is a good reference for either topic?

Qiaochu Yuan points out that it is not at all a coincidence: “The argument that justifies the convergence of geometric series applies in any complete normed field when $|a|<1$, and in the 2-adics this is true when $a=2$.” In the 2-adics, the sequence does indeed converge, and the only thing it can converge to is $1\over1-r$.
