computation SeriesCoefficient

*To*: mathgroup at smc.vnet.net*Subject*: [mg130441] computation SeriesCoefficient*From*: kb <kbru157 at gmail.com>*Date*: Thu, 11 Apr 2013 04:12:48 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net

How is the SeriesCoefficient[f[x],{x,0,n}] calculated for a very large n? The function f[x] has derivatives of all orders. In the case of interest to me f is a rational function with large degree polynomials in numerator and denominator w.o. common factors. The denominator has a simple positive real root smaller in absolute value than all other roots so I can use a simple fraction C_1/(x-a_1) where a_1 is that smallest root and C_1 = Numerator[a_1]/(Denominator)'[a_1], then SeriesCoefficient of f{x} at x=0 is approximately C/a^(n+1). This is pretty good for large n. However, whatever algorithm Mathematica uses is also very good and apparently for all n. Moreover, AbsoluteTiming gives practically 0 sec for both methods for seemingly any n. So, what is the algorithm used? ( I could get better approximations by adding in the next fraction C_2/a_2^(n+1), but then I may run into zeros of higher order or complex zeroes.