AP Calculus BC Unit 10: Infinite Sequences & Series
Study convergence tests, Taylor/Maclaurin series, power series, error bounds with exam-format practice and rubric-based scoring.
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Inside This Unit: The Full Breakdown
Infinite sequences and series are the capstone of BC Calculus. This unit covers convergence tests, Taylor and Maclaurin series, and using series to approximate functions and evaluate limits.
Why it matters
Series questions are the defining feature of the BC exam. At least one full free-response question and many multiple-choice questions test convergence, Taylor polynomials, error bounds, and series manipulation. This is the unit that separates AB from BC.
Key concepts
- A series converges if its sequence of partial sums approaches a finite limit.
- Key convergence tests: nth-term, geometric, p-series, comparison, ratio, integral, and alternating series.
- A Taylor series centered at a represents f(x) as a sum of terms f^(n)(a)/n! * (x-a)^n.
- The Lagrange error bound gives |R_n(x)| <= |f^(n+1)(c)| / (n+1)! * |x-a|^(n+1) for some c between a and x.
Convergence Tests
Start with the nth-term test: if the terms do not approach zero, the series diverges. For series with positive terms, use the comparison test, limit comparison test, ratio test, or integral test. The ratio test is especially effective for series involving factorials or exponentials. The alternating series test applies when terms alternate in sign and decrease in absolute value toward zero. Geometric series with |r| < 1 converge to a/(1-r). P-series with p > 1 converge. Building a systematic approach to choosing the right test is key to exam success.
Taylor and Maclaurin Series
The Taylor series of f centered at a is the sum from n=0 to infinity of f^(n)(a)/n! * (x-a)^n. When a = 0 this is a Maclaurin series. Memorize the Maclaurin series for e^x, sin(x), cos(x), 1/(1-x), and ln(1+x). You can derive new series by substitution, differentiation, or integration of known series. The interval of convergence is found using the ratio test, then checking endpoints separately. Taylor polynomials provide polynomial approximations whose accuracy improves with degree.
Error Bounds and Applications
The Lagrange error bound quantifies the maximum error when approximating f(x) with the nth-degree Taylor polynomial. For alternating series that satisfy the alternating series test, the error is bounded by the absolute value of the first omitted term, which is often easier to apply. The AP exam asks you to determine the minimum degree needed for a Taylor polynomial to approximate a value within a given tolerance. Series can also be used to evaluate limits and integrals that have no closed-form solution.
AP exam tip
Memorize the five key Maclaurin series (e^x, sin x, cos x, 1/(1-x), ln(1+x)) and their intervals of convergence. Most series problems on the exam can be solved by manipulating these known series.
Connections to other units
- Unit 0: Series convergence is fundamentally about limits of partial sums.
- Unit 5: Term-by-term integration of power series uses basic antiderivative rules.
- Unit 6: Power series can represent solutions to differential equations.
