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Interval of Convergence, Radius: Definition, Examples

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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.

interval of convergence

The interval centered about c is the set on which the power series converges is an.

Radius of Convergence

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:
(aR, 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 (aR, a + R).

How to Find the Radius of Convergence

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 an:
root test

Example question: What is the interval of convergence for the series:
root test 1

Solution (perform the root test):

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

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

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:
(aR, 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 xa. The center is “a“.

Ratio Test General Steps

The formula for the test is:
ratio test

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

Step 1: Form a ratio of an + 1/an, 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


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

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Stephanie Glen. "Interval of Convergence, Radius: Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/interval-of-convergence/
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