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.