AP Calculus BC, AP exams, 17–18%

Unit 10: Infinite Sequences and Series

Unit 10 carries a weighting comparable to integration and is the material most unique to BC. It covers convergence tests for series, power series and radius of interval of convergence, Taylor and Maclaurin polynomials, and error bounds including the alternating series and Lagrange error bounds. Choosing the right test quickly is the defining skill.

Position: 10 of 10 units
Published weighting: 17–18% in the official framework
Topics covered: 6 topics in this guide

What is covered

Topics inside this unit.

These topics describe the scope students most often review for this part of AP Calculus BC. Treat them as a study checklist rather than an exhaustive list of what a single exam form can sample.

Sequences and series convergence: The nth-term test, geometric series, and the distinction between a sequence and its series.
Convergence tests: Integral, comparison, limit comparison, ratio, and alternating series tests and when each applies.
Absolute and conditional convergence: Distinguishing the two and understanding what each implies about rearrangement and behavior.
Power series: Radius and interval of convergence, including checking the endpoints separately.
Taylor and Maclaurin series: Constructing series from derivatives, and the standard series for the exponential, sine, cosine, and geometric forms.
Error bounds: The alternating series error bound and the Lagrange error bound for a Taylor polynomial remainder.

Study tips

How to practice Sequences and Series.

Section-specific guidance that follows from how this material is usually tested.

Build a decision tree for convergence tests and practice choosing a test in under fifteen seconds.
Memorize the standard Maclaurin series and practice deriving new ones by substitution, differentiation, and integration.
Practice both error bounds explicitly, since error questions appear regularly and are often skipped in review.
Start series early rather than in the final weeks; this unit carries close to a fifth of the exam.

Where this fits

Unit 10: Infinite Sequences and Series is 10 of 10 units in the AP Calculus BC guide, and it carries a published weighting of 17–18% in the official framework. Use the full guide to balance it against everything else you have to review.

Back to the AP Calculus BC guide Official exam information

Sample Sequences and Series questions

Practice questions, not yet expert-reviewed

Original practice questions written to the published framework for this section. No expert has reviewed them, and they are not taken from any official exam. Try each one before you open the answer.

Question 1Foundational
What is the sum of the infinite geometric series 6 + 2 + 2/3 + 2/9 + ...?
1. A9
2. B8
3. C18
4. DThe series diverges
Show answer
Answer: A. 9
The first term is a = 6 and the common ratio is r = 1/3, so the sum is a/(1 - r) = 6/(2/3) = 9. Dividing by r instead of by 1 - r gives 18, and because |r| < 1 the series certainly converges.
Key insight: A geometric series converges exactly when |r| < 1, and then its sum is a/(1 - r).
Question 2Foundational
Is the following statement true or false: if the terms of a series approach 0, then the series must converge?
1. ATrue
2. BFalse
Show answer
Answer: B. False
The harmonic series, the sum of 1/n, has terms that approach 0 yet diverges, so terms approaching zero is necessary but not sufficient. The nth-term test only works in the other direction: if the terms do not approach 0, the series definitely diverges.
Key insight: The nth-term test can prove divergence but never convergence.
Question 3Intermediate
What is the coefficient of x^3 in the Maclaurin series for e^x?
1. A1/6
2. B1/3
3. C1/2
4. D1
Show answer
Answer: A. 1/6
The Maclaurin series for e^x is the sum of x^n/n!, so the x^3 coefficient is 1/3! = 1/6. Answering 1/3 divides by the exponent rather than by its factorial.
Key insight: Taylor coefficients divide the derivative value by n factorial, not by n.
Question 4Intermediate
What is the radius of convergence of the power series whose nth term is (x - 2)^n/3^n, summed from n = 0 to infinity?
1. A3
2. B1
3. C1/3
4. DThe series converges for all x
Show answer
Answer: A. 3
The series is geometric with ratio (x - 2)/3, so it converges when |x - 2|/3 < 1, that is |x - 2| < 3. The radius of convergence is therefore 3, and the interval is centered at x = 2. Answering 1/3 reports the reciprocal of the radius.
Key insight: Solve |ratio| < 1 for x; the resulting half-width is the radius of convergence.
Question 5Advanced
The alternating series 1 - 1/2 + 1/3 - 1/4 + ... is approximated by the sum of its first four terms. According to the alternating series error bound, what is the maximum possible error?
1. A1/5
2. B1/4
3. C1/6
4. D1/24
Show answer
Answer: A. 1/5
For a convergent alternating series with decreasing terms, the error after n terms is at most the magnitude of the first omitted term. The fifth term is 1/5, so the error bound is 1/5. Using 1/4 quotes the last included term instead of the first omitted one.
Key insight: The alternating series error bound is the size of the first term you left out.

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Unit 10: Infinite Sequences and Series