From: dcons@world.std.com (Denis Constales)
Subject: Re: power-series to primitive function
Date: Mon, 21 Jun 1999 15:06:13 +0200
Newsgroups: sci.math
Keywords: computing infinite sums symbolically

In article <7kl507$dd211@mx2.hrz.uni-essen.de>, hnr00s@power.uni-essen.de (Urs Schreiber) wrote:

> Given any power-series, can one find the finite expression if it
> describes one? For example: I have a+bx+cx^2+... and you see it and
> say: Hey, that's the expansion for artan(ln(x/2)) !!

One has to know the general term of the series as a function of the
summation index and x and possibly other parameters; then some classes of
such functions can be summed exactly, as you indicate, using specific
algorithms (polynomials can be summed using the Euler-Maclaurin formula;
rational functions by splitting into partial fractions and introducing
digamma functions; for stuff with factorials, there's Gosper's algorithm).

The computer algebra package Maple, for instance, provides some examples:

> sum(x^n/n!,n=0..infinity);
                                    exp(x)

> sum(x^n/n,n=1..infinity);
                                  -ln(1 - x)

> collect(sum(x^n/(n^2+5*n+6),n=0..infinity),[ln],factor);
                          (x - 1) ln(1 - x)       x - 2
                        - ----------------- - 1/2 -----
                                  3                 2
                                 x                 x

etc.
--
Denis Constales - dcons@world.std.com - http://world.std.com/~dcons/