AP Calculus BC · Unit 10: Infinite Sequences and Series
10.13 Radius and Interval of Convergence of Power Series
In short
This topic focuses on finding the radius and interval of convergence for power series. You must know that a power series converges either at a single point, on an interval, or for all real numbers. You are expected to use the ratio test to find the radius of convergence.
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What else do you need to know about Radius and Interval of Convergence of Power Series?
Crucially, you must test the endpoints of the open interval individually (using other convergence tests like the alternating series test or p-series test) to determine the exact interval of convergence (open, closed, or half-open). The topic also establishes that term-by-term differentiation and integration of a power series preserve its radius of convergence, though the behavior at the endpoints may change. All content in this topic is exclusive to AP Calculus BC.
Can you do these?
- I can recognize the standard form of a power series.
- I know that a power series converges either at a single point or on an interval of convergence.
- I can test both endpoints of an open interval to determine the full interval of convergence for a power series.
- I know that term-by-term differentiation or integration of a power series does not change its radius of convergence.
How is Radius and Interval of Convergence of Power Series tested on the exam?
On the AP Calculus BC Exam, you are frequently asked to find the interval of convergence for a given power series, often as part of a free-response question. This requires applying the ratio test to find the interior of the interval, followed by explicitly testing both endpoints using appropriate series tests (e.g., alternating series test, harmonic series). A common pitfall is forgetting to test the endpoints or failing to provide sufficient justification for the convergence or divergence at those endpoints. You may also need to identify a re-expression of mathematical information (Skill 2.C), such as recognizing how the radius of convergence applies to a series obtained by differentiating or integrating a known Taylor series.
Free practice questions with full explanations, and instant marking of your free-response answers.
Radius and Interval of Convergence of Power Series: common questions
What do you need to know about Radius and Interval of Convergence of Power Series for AP Calculus BC?
This topic focuses on finding the radius and interval of convergence for power series. You must know that a power series converges either at a single point, on an interval, or for all real numbers. You are expected to use the ratio test to find the radius of convergence.
How is Radius and Interval of Convergence of Power Series tested on the AP Calculus BC exam?
On the AP Calculus BC Exam, you are frequently asked to find the interval of convergence for a given power series, often as part of a free-response question. This requires applying the ratio test to find the interior of the interval, followed by explicitly testing both endpoints using appropriate series tests (e.g., alternating series test, harmonic series).
How do I know I have mastered Radius and Interval of Convergence of Power Series?
Check you can do each of these: recognize the standard form of a power series; I know that a power series converges either at a single point or on an interval of convergence; test both endpoints of an open interval to determine the full interval of convergence for a power series; I know that term-by-term differentiation or integration of a power series does not change its radius of convergence.
Where does Radius and Interval of Convergence of Power Series fit in AP Calculus BC?
Radius and Interval of Convergence of Power Series is topic 10.13 in Unit 10: Infinite Sequences and Series of AP Calculus BC.
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