Asymptotia

*To*: mathgroup*Subject*: Asymptotia*Date*: Mon Mar 20 14:53:19 1989

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.