You may remember from geometric series that

for appropriate values of *r*. Similarly, this tells us from a power
series
perspective that

when *x* is between -1 and 1. So, the function 1/(1-*x)* can
be represented
as a power series for part of its domain. In similar ways, other functions
can be represented by power series.

Differentiation and integration are useful techniques for finding power series representations of functions. Differentiation and integration of power series works in a way very similar to handling polynomials: look at the series term by term. For instance, look at the power series

with radius of convergence *R*, and define *f*(*x*) on
the interval (*a*-*R*,*a*+*R*) by setting it equal to
the series.
Then,

and

.

The radii of convergence of these power series will both be *R*,
the same
as the original function.

For instance, suppose you were interested in finding the power series representation of

We can find the power representation of this function like so:

,

or

.

,

where *y*=*x*/3, so

.

Thus,

.

Copyright © 1996 Department of Mathematics, Oregon State University

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