Calculus BC · Unit 10: Infinite Sequences and Series · 14 min read · Updated 2026-05-10
Alternating series test for convergence — AP Calculus BC
AP Calculus BC · Unit 10: Infinite Sequences and Series · 14 min read
1. The Alternating Series Test: Definition and Conditions★★☆☆☆⏱ 4 min
An alternating series is any infinite series where consecutive terms alternate in sign. The Alternating Series Test (AST, also called Leibniz's Test) is the standard convergence test for this series type, and it is frequently tested on AP Calculus BC, appearing in both multiple-choice and free-response sections.
The Alternating Series Test states that an alternating series **converges** if both of the following conditions hold for all sufficiently large $n$ (after some finite starting index $N$):
The sequence of positive terms $\{a_n\}$ is eventually strictly decreasing: $a_{n+1} < a_n$ for all $n > N$
The limit of the positive terms approaches zero: $\lim_{n \to \infty} a_n = 0$
2. Alternating Series Remainder Estimation★★★☆☆⏱ 4 min
After confirming an alternating series converges via AST, the AP exam often asks to bound the error (called the remainder) when approximating the total sum $S$ with a finite partial sum $S_n$. The Alternating Series Remainder Theorem gives a simple, exam-friendly bound for this error.
3. Classifying Convergence: Absolute vs Conditional★★★☆☆⏱ 4 min
After confirming an alternating series converges, the AP exam almost always asks to classify its convergence as either absolute or conditional. This distinction is especially important for finding the interval of convergence of power series, where endpoints are often conditionally convergent.
4. Exam-Style Concept Check★★★★☆⏱ 2 min
Common Pitfalls
Why: Students confuse the signed overall terms with the $a_n$ defined in AST, which must be the positive magnitude of each term.
Why: Students forget convergence only depends on the behavior of terms for large $n$, and early terms do not affect convergence.
Why: Students memorize the decreasing condition but forget the limit condition is required for convergence.
Why: The bound only applies to convergent alternating series, not divergent alternating series or positive-term series.
Why: Students confuse convergence of the alternating series with convergence of the series of absolute values, which is required for absolute convergence.