AP® Calculus BC · Unit 10 · Topic 10.13
Radius of Convergence of a Power Series
The radius of convergence is the distance from a power series' center within which the series definitely converges; you find it by applying the ratio test to the general term and solving the resulting inequality for x.
A power series is built from powers of (x - c), each with its own coefficient. Near the center c, the terms shrink fast enough that the sum settles down. Far from c, the terms grow and the sum blows up. The radius of convergence, R, is the exact cutoff distance: the series converges for all x within R of c, and diverges for all x farther than R away. At exactly R away, on the two endpoints, anything can happen, which is why those points need separate checking.
To find R, apply the ratio test to the general term an. Compute the limit as n approaches infinity of |a_(n+1)/an|, which usually simplifies to some expression times |x - c|. Convergence requires this limit to be less than 1, which turns into an inequality like |x - c| < R. Whatever number sits on the right after isolating |x - c| is the radius. For sum of (x-2)n / (n times 3n), the ratio test gives |x-2|/3 < 1, so R = 3. For sum of n! xn, the ratio of consecutive terms gives (n+1)|x|, which only stays below 1 when x = 0, so R = 0: the series converges at a single point.
A function can be defined everywhere, but its power series representation often is not. The series is a local approximation tool, and it only matches the function inside its radius of convergence. Once R is found, the real work shifts to the two boundary points x = c - R and x = c + R, where the ratio test gives no answer and you must plug in directly and test the resulting numerical series by hand, which is exactly the next step: finding the interval of convergence.
Key terms
- Radius of convergence
- The distance R from the center c such that the power series converges for |x - c| < R and diverges for |x - c| > R.
- Ratio test
- A test that examines the limit of |a_(n+1)/an| to determine where a series converges absolutely.
- Center
- The value c around which the power series sum of an (x-c)n is built.
- Interval of convergence
- The full set of x-values, including any convergent endpoints, where the power series converges.
Watch it explained
- How it worksHow does the ratio test turn a power series into an inequality that x must satisfy?
- Worked exampleFind the radius of convergence for the sum of (x - 2)n divided by n times 3n.
- Worked exampleFind the radius of convergence for the sum of n factorial times xn.
- Common mistakeWhy doesn't a power series converge for every x just because the function is defined there?
- Ties togetherHow does finding the radius of convergence set up checking the two endpoints by hand for the interval of convergence?
Check yourself
End of topic.