Calculus BC · Unit 10: Infinite Sequences and Series · 14 min read · Updated 2026-05-11
Ratio Test for Convergence — AP Calculus BC
AP Calculus BC · Unit 10: Infinite Sequences and Series · 14 min read
1. Core Concepts of the Ratio Test★★☆☆☆⏱ 4 min
The Ratio Test (also called d'Alembert's Ratio Test) is a convergence test for infinite series $\sum_{n=1}^\infty a_n$, most useful when series terms contain factorials, exponential functions, or powers of $n$ that simplify cleanly when taking the ratio of consecutive terms. This topic is tested on both multiple-choice and free-response sections of the AP Calculus BC exam.
2. Inconclusive Cases★★★☆☆⏱ 3 min
When $L = 1$, the ratio test gives no information about convergence, and this is a common testing point on the AP exam. The ratio test only captures exponential growth rates of terms; when growth is polynomial, the limit $L$ will always equal 1, regardless of whether the series converges or diverges. Whenever you calculate $L=1$, you must switch to another appropriate test.
3. Applying the Ratio Test to Power Series★★★☆☆⏱ 4 min
The most common high-stakes use of the ratio test on the AP Calculus BC exam is finding the radius and interval of convergence for a power series of the form $\sum_{n=0}^\infty c_n (x-a)^n$. Convergence of a power series depends on the value of $x$, and the ratio test is ideal here because the $(x-a)$ term simplifies cleanly when taking the ratio of consecutive terms.
Write $a_n = c_n (x-a)^n$, then set up the ratio $\left| \frac{a_{n+1}}{a_n} \right|$
Separate terms involving $x$ from terms involving $n$