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?

As for references, Hardy wrote a book, Divergent Series, which seems to be available for download on the web. There are several books about the $p$-adics; Koblitz, $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions; Bachman, Introduction to $p$-adic Numbers and Valuation Theory; Gouvea, $p$-adic Numbers. There's also an article I wrote with Alf van der Poorten, Some problems concerning recurrence sequences, Amer. Math. Monthly 102 (1995) 698-705,  MR 97a:11029. 
