AP® Calculus BC · Unit 10 · Topic 10.13
Interval of Convergence and Testing the Endpoints
The ratio test gives you a radius of convergence but says nothing about the two endpoints, so you plug each one in separately and test the resulting number series on its own.
A power series sum of an(x-c)n converges on some interval centered at c. The ratio test finds the radius R by looking at the limit of |an+1/an| times |x-c|, then solving where that limit is less than 1. But right at the two endpoints x = c - R and x = c + R, that limit equals exactly 1. The ratio test's verdict there is always "inconclusive" — it was built to compare growth rates, and at the edge the series terms and the comparison ratio tie exactly. So the test goes silent, and you're left with two ordinary numerical series, one per endpoint, that need a separate test: p-series, alternating series, comparison, or divergence test.
This is why an interval of convergence can look different on each side: closed, open, or half-open, with no pattern forcing symmetry. Plugging in the left endpoint might produce an alternating series that converges conditionally, while the right endpoint produces a positive p-series that diverges. The algebra of substitution is symmetric, but convergence behavior of the resulting constant series is not — each endpoint earns its answer independently by its own test, not by a rule about endpoints in general.
This step is where the whole unit reconnects: the endpoint series are exactly the p-series and alternating series you studied earlier, now reappearing as the last piece of a power series problem. Getting the interval of convergence exactly right, including which brackets are curved and which are square, matters directly for the next topic, where you use these series to represent and manipulate functions only where they actually converge.
Key terms
- Radius of convergence
- The distance R from the center within which the power series is guaranteed to converge absolutely.
- Interval of convergence
- The full set of x-values where the power series converges, including whichever endpoints pass their own test.
- p-series
- A series of the form sum of 1/np, which converges if p > 1 and diverges if p ≤ 1.
- Alternating series test
- A test showing that an alternating series converges if its terms shrink to zero and decrease in absolute value.
- Conditional convergence
- When a series converges but the series of its absolute values diverges.
Watch it explained
- How it worksHow does the ratio test go silent exactly at the endpoints and force a different test?
- Worked exampleFind the full interval of convergence for the sum of (x + 1)n divided by √n.
- Worked exampleFind the interval of convergence for the sum of xn divided by n2, endpoints included.
- Common mistakeWhy isn't the interval of convergence always closed at both ends or open at both ends?
- Ties togetherHow does testing these endpoints call back on the alternating series and p-series tests from earlier in the unit?
Check yourself
End of topic.