Unit 10 Practice Test: Infinite Sequences and Series (AP Calculus BC)

The Most Challenging BC Unit: Sequences and Series

Unit 10 is exclusively BC content and is widely considered the most difficult unit in the AP Calculus BC curriculum. It introduces infinite sequences and series, requiring students to master multiple convergence tests, construct power series representations, determine intervals of convergence, and apply Taylor and Maclaurin series to approximate functions and bound errors. This unit often has the greatest impact on final BC scores and deserves dedicated, structured preparation.

Key Topics in Unit 10

Sequences and Convergence

An infinite sequence converges if its terms approach a finite limit. Students must evaluate limits of sequences, often using L'Hôpital's Rule or standard limit techniques from Unit 1. Recognizing divergent sequences — those that grow without bound or oscillate — is equally important.

Series Convergence and Divergence Tests

Determining whether an infinite series converges requires selecting and correctly applying the appropriate test. BC students must know all of the following:

• Divergence Test (nth Term Test): If the terms do not approach zero, the series diverges. Note that the converse is not true.
• Geometric Series: Converges when |r| < 1; sum = a/(1−r).
• p-Series: Converges when p > 1.
• Integral Test: Connects series convergence to the convergence of a corresponding improper integral.
• Comparison Test: Compares a series to a known convergent or divergent benchmark series.
• Limit Comparison Test: Uses the ratio of terms to compare asymptotic behavior.
• Ratio Test: Most effective for series with factorials or exponentials; determines convergence based on the limit of the ratio of consecutive terms.
• Alternating Series Test: Applies to alternating series when terms decrease in absolute value and approach zero.

Power Series and Radius of Convergence

A power series is an infinite sum involving powers of (x − c). The radius of convergence R is found using the Ratio Test. The interval of convergence is determined by testing the endpoints separately — a step that many students omit and that is explicitly tested on the BC exam.

Taylor and Maclaurin Series

Taylor series represent functions as infinite polynomial sums centered at a given point. Maclaurin series are Taylor series centered at zero. BC students must memorize or derive the standard Maclaurin series for e^x, sin x, cos x, and 1/(1−x), and be able to adapt them through substitution, differentiation, or integration to represent related functions. FRQ questions frequently ask students to write a Taylor series for a function and use it to approximate a value or integral.

Error Bounds

Two error bound methods appear on BC exams:

• Alternating Series Error Bound: For an alternating series, the absolute error is bounded by the absolute value of the first omitted term.
• Lagrange Error Bound: Bounds the error of a Taylor polynomial approximation using the maximum value of the next derivative on the interval.

Selecting the correct error bound method for a given problem — and setting it up properly — is a graded skill on BC FRQs.

Most Common Convergence Test Errors

The most frequent student errors in Unit 10 are: applying the Divergence Test to conclude convergence (it cannot), failing to check endpoints when stating the interval of convergence, misapplying the Ratio Test to series for which it is inconclusive, and confusing the Alternating Series Error Bound with the Lagrange Error Bound. Consistent practice with test selection — not just execution — is essential.

How to Approach Taylor Series FRQs

Taylor series FRQ questions typically unfold in parts: writing the first several terms of a series, finding the general term, determining the interval of convergence, using the series to approximate a value, and bounding the error. Students who skip the general term or forget to test endpoints in the interval of convergence lose points across multiple parts of the same question. Practicing the complete FRQ arc — not just individual steps — is the most effective preparation strategy.

Unit 10 Practice Test Coverage

Practice questions include sequence convergence analysis, convergence test selection and application across all BC-required tests, power series radius and interval of convergence problems including endpoint testing, Taylor and Maclaurin series construction and adaptation, function approximation using Taylor polynomials, and error bound problems using both alternating series and Lagrange methods. Questions are structured in multi-part AP Calculus BC FRQ format to reflect the full arc of series questions on the exam.