# Understanding the Interval of Convergence

Unlike geometric series and *p*-series, a power series often converges or diverges based on its *x *value. This leads to a new concept when dealing with power series: the interval of convergence.

The *interval of convergence* for a power series is the set of *x* values for which that series converges.

## The interval of convergence is never empty

Every power series converges for some value of *x**.* That is, the interval of convergence for a power series is never the empty set.

Although this fact has useful implications, it’s actually pretty much a no-brainer. For example, take a look at the following power series:

When *x* = 0, this series evaluates to 1 + 0 + 0 + 0 + ..., so it obviously converges to 1. Similarly, take a peek at this power series:

This time, when *x* = –5, the series converges to 0, just as trivially as the last example.

Note that in both of these examples, the series converges trivially at *x* = *a* for a power series centered at *a*.

## Three possibilities for the interval of convergence

Three possibilities exist for the interval of convergence of any power series:

The series converges only when

*x*=*a**.*The series converges on some interval (open or closed at either end) centered at

*a**.*The series converges for all real values of

*x**.*

For example, suppose that you want to find the interval of convergence for:

This power series is centered at 0, so it converges when *x* = 0. Using the ratio test, you can find out whether it converges for any other values of *x**.* To start out, set up the following limit:

To evaluate this limit, start out by canceling* x** ^{n}* in the numerator and denominator:

Next, distribute to remove the parentheses in the numerator:

As it stands, this limit is of the form

so apply L’Hopital’s Rule, differentiating over the variable *n**:*

From this result, the ratio test tells you that the series:

Converges when –1 <

*x*< 1Diverges when

*x*< –1 and*x*> 1May converge or diverge when

*x*= 1 and*x*= –1

Fortunately, it’s easy to see what happens in these two remaining cases. Here’s what the series looks like when *x* = 1:

Clearly, the series diverges. Similarly, here’s what it looks like when *x* = –1:

This alternating series swings wildly between negative and positive values, so it also diverges.

As a final example, suppose that you want to find the interval of convergence for the following series:

This series is centered at 0, so it converges when *x* = 0. The real question is whether it converges for other values of *x**.* Because this is an alternating series, you apply the ratio test to the positive version of it to see whether you can show that it’s absolutely convergent:

First off, you want to simplify this a bit:

Next, you expand out the exponents and factorials:

At this point, a lot of canceling is possible:

This time, the limit falls between –1 and 1 for all values of *x**.* This result tells you that the series converges absolutely for all values of *x**,* so the alternating series also converges for all values of *x**.*