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?

This is not so much an answer as a related reference. I wrote a short expository note "Divergence is not the fault of the series," Pi Mu Epsilon Journal, 8, no. 9, 588-589, that discusses this idea and its relation to 2's complement arithmetic for computers.
