A power series is a summation, from i=0 to i=n, of ai x^i where x is a complex variable.
a0 + a1 x + a2 x^2 + a3 ^3 + ... + ai x^i + ... an x^n.
The infinite power series is the limit as n increases without bound. It can be shown that the infinitie power series converges in a circle of radius r; that is, it converges for all x whose absolute value is less that r, diverges for all x whose absolute value is greater than r, and may either converge or diverge for specific values of x whose absoulte value is equal to r. The radius of convergence r is the distance to the closest singularity of the function represented by the infinite power series.
The actual computation of the power series is performed best by factoring
a0 (1 + x a1/a0 (1 + x a2/a1 (1 + x a3/a2 (1 + x a4/a3 (1 + ...)))))
and evaluating from the inside out. This factored form requires fewer multiplications and has less round-off error. Of course, suitable modifications have to be perfomed if any of the coefficients are zero.
Usually, the power series is obtained as the Taylor series of the function, about the point p. Each coefficient ai then is the i-th derivative of the function., evaluated at the point p, divided by i!. Since the derivatives are unique, it follows that the same infinite power series will result from any method of derivation.
We will display the first several terms, the general term ai x^i, and (for the purpose of the factorization) the ratio of the (n+1)-th term to the n-th term x a(n+1) / an.
Copywrite (c) 1997 R. I. 'Sciibor-Marchocki last modified on Tuesday 06-th May 1997.