Introduction to Power Series

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 nth 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 nth root of the absolute value of c[n] is K, then the radius of convergence is 1/K.

If you have questions or comments, don't hestitate to contact us.