Finding Intervals of Convergence and Radii of Convergence with the Ratio Test
By Sarah Breslow, Maria Bromme, and Dahlia Goldfeld
When you write a Taylor polynomial approximation of a function, for example:
and graph it, there is an interval on the graph where the Taylor polynomial and the graph of the actual function come really close together. This is where the Taylor polynomial converges, and it is known as the interval of convergence. The radius of convergence is half the interval of convergence.
For example: Let’s say an interval of convergence is (-3,5). The absolute value of –3 and 5, or |-3 + 5| equals 8. Half of 8 is 4, and so 4 is the radius of convergence. Got it? Awesome!
To find the interval of convergence for any series, we use this really cool thing called the ratio test! Basically, you do some arithmetic, take a limit, solve an inequality, and you have your interval of convergence.
Note: After finding you interval of convergence, you still need to test the boundaries to see if the series converges at the end points. (You would test, in our previous example, the endpoints –3 and 5 in your boundary tests). Our study guide doesn’t cover boundary tests; make sure to check out your CARPING study guide for that.
And finally, what you’ve been waiting for, the most important part of all– the ratio test!
For any series: an =
The interval of convergence exists where:
Example 1: ex =
Find the interval and radius of convergence.
Therefore, the interval of convergence is (-1,1), and the radius of convergence is 1.
(Now you would do carping to test the boundaries: -1 and 1, but as we said before, you have to go to your CARPING guide to finish this problem).
Example 2:
Find the interval and radius of convergence.
Thus, the interval of convergence is (-1,9)
Thus, the radius of convergence is 5.
(Again, you have to do carping for the boundaries: -1 and 9, to completely finish this problem).