1 + 2 + 4 + 8 + ... = -1

Not really. 0 =/= 1 + 2(S), so you cannot say S = -1

The sum S doesn't converge, so you're essentially saying an one aleph-null infinity is the same as another aleph-null infinity. Big deal.

It does converge using the "p-adic norm" - i.e., if you change the way in which you measure the difference between two numbers.

If you take a rational number n and write it in the form
[LATEX]\displaystyle n=\frac{2^k r}{s}[/LATEX], where r and s are odd integers, and where k is a positive or negative integer, then the 2-adic norm of n is [LATEX]\|n\|_2 =2^{-k}[/LATEX]. The norm is a measure of how big a number is.

So, for instance [LATEX]\displaystyle \| 3 \|_2 = 1; \|4\|_2 = \frac{1}{2}; \left\|\frac{7}{12}\right\|_2 = 4[/LATEX]. It's a strange notion: 3 is a "bigger" number than 4, and both are "smaller" than 7/12!

We can show that the sum in the OP converges in the 2-adic norm to -1. We look at the size of the different between the first n terms in the sum, and the supposed limit -1:
[LATEX] \displaystyle
\| S_n - (-1) \| = \left\| \sum_{k=0}^n2^k -(-1) \right\| = \| 2^{n+1} -1 +1 \| = \| 2^{n+1} \| = \frac{1}{2^{n+1}} \rightarrow 0
[/LATEX] as [LATEX]n \rightarrow \infty [/LATEX]. So in the 2-adic norm, [LATEX]\displaystyle \lim_{n\rightarrow \infty} \sum_{k=0}^n 2^k =-1[/LATEX] .

The p-adic norm actually has some pretty nice "applications" - it's not a triviality! Naturally, the Wikipedia article doesn't mention these...but I've seen some, and they are nice.

Last year at the Irish Student Math Conference a professor from NUI Maynooth proved a theorem about binomial co-efficients using p-adic norms. Fix a rational number a. Then if the denominator of the binomial coefficient [LATEX]\displaystyle {a \choose k}= \frac{a(a-1)...(a-n+1)}{k!}[/LATEX] (for k an integer) has a given prime p in it, then a also has the same p in its denominator.

The approach was to show that the function [LATEX]\displaystyle a \mapsto {a \choose k}[/LATEX] is continuous with respect to the p-adic norm, and that hence the inverse image of some closed set was closed, and then something about the inverse image only containing numbers with primes p in their denominator. It was a nice mix of arithmetic/algebra/kind of number theory with analysis.