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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
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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.
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