AP Calculus BC
Radius vs Interval of Convergence
The radius of convergence is a single number, the distance from the centre to where convergence stops, found with the ratio test. The interval of convergence is the full set of inputs where the series converges, which requires testing each endpoint separately.
Radius
Use when: The question asks only how far the series converges from its centre.
Interval
Use when: The question asks for all values of the variable where the series converges.
Side by side
| Radius | Interval | |
|---|---|---|
| Answer type | A number | An interval with brackets |
| Found with | The ratio test | The ratio test plus endpoint tests |
| Endpoints | Not considered | Tested one at a time |
| Ratio test at the endpoint | Not applicable | Always inconclusive there |
The ratio test gives strict inequality, so it is silent exactly at the endpoints. That is not a flaw to work around; it is why endpoint testing is a separate step with a different test, usually the alternating series test on one side and a p-series comparison on the other.
All four bracket combinations are possible, and the two endpoints very often behave differently, so each must be checked on its own.
The mistake
Reporting the radius and stopping when the question asked for the interval, or assuming both endpoints behave the same way.
Frequently asked questions
Can the radius be zero or infinite?
Yes. A radius of zero means the series converges only at its centre, and an infinite radius means it converges everywhere, as the series for does.
In the CED: Unit 10: Infinite Sequences and Series (BC)