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2019-02-18, 17:55   #7
Dr Sardonicus

Feb 2017
Nowhere

11·317 Posts

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 2-adic valuation! Under this "non-Archimedian" 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 p-adic 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

Last fiddled with by Dr Sardonicus on 2019-02-18 at 18:02