From: David Ullrich Newsgroups: sci.math Subject: Re: Help wanted about series covergence Date: 27 Oct 1995 19:32:57 GMT Shubin Liu wrote: > >Hi, all: > >I have following series: > > Q(x) = A1/x + A2/(2*x^2) + A3/(3!*x^3) + ... > >where A1, A2, A3 ... are expansion coefficients. I hope that it will >converge even when x approaches zero. This may be possible when certain >conditions are superposed on these coefficients. > >My questions are: (1) is this possible? Whether this is possible depends on exactly what you mean by "converge when x approaches zero". It's certainly possible for the series to converge for every non-zero x, so maybe the question is whether the function Q(x) can have a limit as x tends to 0. In fact Q can have a limit as x tends to zero (if x is restricted to be real - it's impossible for the limit to exist as x tends to zero in the complex plane except in the trivial case where all the coefficients vanish). Don't know about general conditions, but here's an example showing it's possible: Let An=(-1)^n. You know that e^x tends to 0 as x tends to minus infinity - now if you look at the power series for e^x and replace x by -1/x you get what I said. Well, that Q has a one-sided limit at the origin; you can get a two-sided limit by a similar application of the fact that e^(-x^2) tends to 0 as x tends to plus or minus infinity. But (assuming that not all the An vanish) Q cannot have even a one-sided limit at 0 if unless infinitely many of the An are non-zero (if all but finitely many An vanish but they don't all vanish then Q must blow up at the origin). And if infinitely many of the An are non-zero then Q , regarded as a function of a complex variable, has an essential singularity at the origin, which implies that its behavior is very bad in most directions - the fact that it's possible to have a limit along the real axis is sort of delicate. -- David Ullrich Don't you guys find it tedious typing the same thing after your signature each time you post something? I know I do, but when in Rome...