[FOM] Hardy's "Divergent series"

Gabriel Stolzenberg gstolzen at math.bu.edu
Mon Sep 24 12:09:55 EDT 2007

   On Saturday, September 22, Joe Shipman asked:

> 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?

   Have you consulted Hardy's "Divergent Series"?  (For analytic
continuation, see pp.186-191.)

   I don't remember if Hardy discusses it, but one nice feature of
some of the methods for "converting" divergent series into convergent
ones is that, in some cases, they also convert a slowly converging
series, like the alternating harmonic series, into a rapidly converging
one, like the power series expansion of log(1 - x) at x = 1/2.  (But
we can't iterate.  They convert rapidly converging series into slowly
converging ones.)

   Gabriel Stolzenberg

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