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*Sequences and Series*starting page - Representing functions as power series
- Taylor Series
- Error Bounds using Taylor Polynomials

A power series is a series of the form

where *x* is a variable and the *c*[*n*] are constants
called the coefficients
of the series. We can define the sum of the series as a function

with domain the set of all *x* for which the series converges.

More generally, a series of the form

is called a power series in (*x*-*a*) or a power
series at
*a*. So, the question
becomes "when does the power series converge?" Any of the series tests are
available for use, but most often the
Ratio Test is used. It
tells us
that
the series converges when the limit of the ratio of the *n*+1st term
to the
*n*th term is less than one in absolute value, and diverges when the
limit
is greater than one in absolute value. In general, this boils down to

.

When this limit is between -1 and 1, the series converges.

There are only three possibilities for how this series can converge:

- The series only converges at
*x*=*a* - The series converges for all
*x* - There is some positive number
*R*such that the series converges for |*x*-*a*|<*R*and diverges for |*x*-*a*|>*R*.

In the third case, *R* is called the radius of
convergence.
Note that the special cases of |*x*-*a*|=*R*
need to be checked separately. If the series only converges at *a*,
we say the radius of convergence is zero, and if it converges everywhere,
we say the radius of convergence is infinite.

For example, look at the power series

Using the ratio test, we find that

so the series converges when *x* is between -1 and 1. If *x*=1,
then we get

which diverges, since it is the
harmonic
series. If *x*=-1, then we get

which converges, by the
Alternating Series
Test.
So, the power series above converges for *x* in [-1,1).

One fact that may occasionally be helpful for finding the radius of
convergence: if the limit
of the
*n*th root of the absolute value of *c*[*n*] is *K*,
then the radius of convergence
is 1/*K*.

- Return to the
*Sequences and Series*starting page - Representing functions as power series
- Taylor Series
- Error Bounds using Taylor Polynomials

Copyright © 1996 Department of Mathematics, Oregon State University

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