Quote:
Originally Posted by LaurV
Hm... are you sure? :wondering:
I thought 1+1+1+1+....= 1/2

Hmm. I've seen the alternating series 1  1 + 1  1+ ... or
Grandi's series being given the value 1/2, which answer can be obtained using
Cesaro summation. But all terms +1, I don't know a reasonable way to assign a sum.
I do however, know a "standard" way to sum the geometric series
1 + 2 + 4 + 8 + ...
and get 1, the sum given by blindly applying the usual formula for a geometric series
1 + x + x^2 + x^3 + ... = 1/(1  x).
It's perfectly valid 
if you use the 2adic valuation! Under this "nonArchimedian" valuation of the rationals, 2 = 1/2, so positive integer powers of 2 are "small," and the series is convergent. However,
negative integer powers of 2 are now "large," so the geometric series
1 + 1/2 + 1/4 + ...
is now divergent!
EDIT: It suddenly occurs to me, under the padic valuation with p = 1/p, the series whose terms are all 1 is not only bounded, but has a subsequence of partial sums tending to 0. Every partial sum of a multiple of p terms is "small," of a multiple of p^2 terms even smaller, and so on. Alas, I'm too lazy to try and tease a sum out of this