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MathGroup Archive 1989

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Asymptotia

  • To: mathgroup at yoda
  • Subject: Asymptotia
  • From: "Charles W. Clark, NIST, (301) 975-3738" <CLARK at ENH.NBS.GOV>
  • Date: Mon, 20 Mar 89 14:51:48 CST

I am sure that M'ica does its sums in a very clever way, but it seems
to take some chances at the margin.

Consider the classical case of a divergent series, the asymptotic
expansion for the exponential integral (Whittaker and Watson,
chapter VIII): n! (-x)^n. M'ica (running on a Mac IIx
, local kernel,
default settings) returns a finite result without warning for 
surprisingly large values of x:

x = 0.04934
Sum[n! (-x)^n, {n, 0, Infinity}] //N
0.954923

though, to be entirely fair, it does spot trouble for larger values:

x = 0.05
Sum[n! (-x)^n, {n, 0, Infinity}] //N
NLimit::dubious: The result from NLimit may be completely wrong.
0.954371

If the results of the ratio test are looked at properly (i.e.
nth ratio = (n+1) x, grows indefinitely large with n), this
series is seen to be divergent even at the smallest values of
x.  This sort of test picks out the divergent series that most
frequently occur in asymptotic expansions, and I wonder why it
has not been implemented.





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