Call . Cancelling off the very first term by subtracting 1 gives so or .
Call . Add the series and together term by term to get . Since , or .
The remarkable thing, of course, is that you get the same answer if you do it "algebraically" like this as if you do it like Opalg does it (analytically, generating funtions).
1 ? 2 + 3 ? 4 + · · · - Wikipedia, the free encyclopedia
I don't know enough math to understand the "proofs" in that article. Can anyone explain the number manipulation method? It seems like the proof that requires the least amount of math knowledge to understand.
EDIT: I can see how this would be 1/4 in the formal power sense (which is what maddas was saying), but this sub forum is not that advanced.
The value of the series depends on the method used to sum it. Everyone knows that the series diverges in the traditional sense (aka. the partial sums are eventually graeater in magnitude than any number). The series is Abel summable to 1/4. The wikipedia article the OP linked discusses other summability methods under which it converges. In some sense, any "reasonable" summation method which sums this series must give 1/4. Hardy has a book on divergent series for anyone interested in these sums and their applications.
(Also, Wilmer, you have wrong series; the signs alternate every term, not every two terms. The partial sums are actually 0,1,-1,2,-2,3,-3,...)