[FOM] Divergent series

Francois G. Dorais dorais at math.cornell.edu
Sat Sep 22 18:46:17 EDT 2007

joeshipman at aol.com wrote:
> What is the most general way known to consistently assign sums to 
> divergent series a0 + a1 + a2 + ... ? "Analytic continuation of a0 + 
> a1x + a2x^2 + ...to x=1 when the answer is unique" obviously is 
> consistent, but are there other methods consistent with this which 
> extend to situations where the power series has a 0 radius of 
> convergence and so cannot be analytically continued?

You may be interested in Banach limits (see
http://en.wikipedia.org/wiki/Banach_limit for a brief summary).  I
believe the "analytic continuation method", as you used for 1+2+3+...,
and Abel summation, as you outlined above, are both of that type, at
least when appropriately formulated.

François G. Dorais
Department of Mathematics
Cornell University

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