Interval of Convergence, Radius: Definition, Examples

Calculus Definitions >

The interval of convergence is a set of x-values on which a power series converges. In other words, it’s the interval of x-values that you can plug in to make a convergent series. It’s possible for this interval to include all of the values in a series, a limited range of x-values, or just a single x-value at the center.

Within this interval, the series has both absolutely convergence and uniform convergence.

Radius of Convergence

For real-valued functions, the radius of convergence is half the length of the interval of convergence. For example, let’s say you had the interval (b, c). The radius of convergence will be R = (c – b) / 2. Two extremes are possible:

• The radius of convergence can be zero, which will result in an interval of convergence with a single point, a (the interval of convergence is never empty).
• Or, for power series which is convergent for all x-values, the radius of convergence is +∞.

The anchor point a is always the center of the interval of convergence. For real numbers of x, the interval is a line segment; for complex numbers, the interval will be a circle. If the radius of convergence is R, then the interval of convergence includes the open interval:
(a – R, a + R).
For example, if the radius of convergence is 1/N, then the interval of convergence includes (a – 1/N, a + 1/N); In order to find the interval of convergence, find the radius of convergence first then insert the result into (a – R, a + R).

For the radius of convergence for complex-valued functions, see: Circle of Convergence

How to Find the Radius of Convergence

Watch the video for an example of how to find the radius of convergence finding the root test:

Can’t see the video? Click here.

The ratio test or the root test can be used. This first example uses the root test, which works by taking the limit of a series a_n:

Example question: What is the interval of convergence for the series:

Solution (perform the root test):

Step 1: Plug the series into the formula for the root test:

Step 2: Set the limit as an equality less than 1 (for convergence):

Step 3: Solve for x:

The Radius of Convergence is 1 (from the right side of the inequality).

Step 4: Plug your Step 3 answer for R into the interval of convergence formula:
(a – R, a + R) = (5 – 1, 5 + 1) = (4, 6).

*For a power series, the center is defined in the terms. Look for part of a general term in the series that looks like x – a. The center is “a“.

Ratio Test General Steps

The formula for the test is:

The basic steps for using the ratio test to find the radius of convergence:

Step 1: Form a ratio of a_n + 1/a_n, then simplify.

Step 2: Take the absolute value of the ratio and the limit as n → ∞

Step 3: Use the table below to find R.

Result from Step 2:		R
Zero		Infinite (i.e. convergence for all x-values)
N · |x – a| (N is a finite, positive number)		R = 1/N
Infinity		Zero (i.e. convergence at only x = a)

A Note About Endpoints

The steps above show you how to find the interval of convergence, but they don’t tell you if the endpoints of the series are inside that interval. To find that out, plug the endpoints into the power series one at a time. Then use a convergence test to figure out if the infinite series you just created (by plugging in one of the endpoints) converges:

• If it does converge, then the endpoints are in the interval of convergence.
• If it doesn’t converge, the endpoints are not in the interval.

Don’t use the ratio test or the root test for testing the endpoints: it tends to not work well, and will likely give you inconclusive results.

References

Belding, D. & Mitchell, K. (2008). Foundations of Analysis. Courier Corporation.
Krantz, S. (2004). A Handbook of Real Variables: With Applications to Differential Equations and Fourier Analysis. Springer Science and Business Media.
Winter, D. (2008). Recitation Handout 17: Radius and Interval of Convergence. Retrieved February 2, 2020 from: http://www.math.cmu.edu/~amanita/math122/handouts/m122_f08_rhandout17.pdf

------------------------------------------------------------------------------

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!