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?

A counterexample may be interesting. Consider the sequence

$$ a_n = \frac{n!}{n! + 1} $$

In the real numbers, we obviously have

$$ \lim_{n \to +\infty} a_n = 1, $$

but in every $p$-adic field, we have

$$ \lim_{n \to +\infty} a_n = 0, $$

so you should be wary of the idea of trying to sum a series of real numbers by transplanting it to the $p$-adics.

One thing you can consider is $\mathbb{Q}((x))$, the field of rational (formal) Laurent series. In this field, you have an identity

$$ \sum_{n=0}^{+\infty} x^n = \frac{1}{1-x}. $$

There is no issue of convergence or anything here; you just check that multiplying the left hand side by $1-x$ gives you $1$.

There is a subfield of $\mathbb{Q}((x))$ that consists of only those Laurent series for which replacing $x$ by $2$ yields a convergent $2$-adic sum. Evaluation at $2$ then becomes a field homomorphism to the $2$-adic numbers. Since $\sum_{n=0}^{+\infty} x^n$ is in that subfield, its image in $\mathbb{Q}_2$ must be the same as the image of $1/(1-x)$: i.e. $-1$.

Wikipedia has a page on divergent series which talks about "summation methods". You may find this another useful starting point.

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.