Radius and interval of convergence of power series — AP Calculus BC
1. Core Definitions of Power Series Convergence ★★☆☆☆ ⏱ 3 min
A power series is an infinite series centered at constant $a$, with constant coefficients $c_n$. For any input $x$, the series reduces to a numerical infinite series that either converges or diverges. This topic makes up 17-18% of the total AP Calculus BC exam score, and appears in both multiple-choice and free-response sections, almost always paired with other series topics like Taylor series.
2. Finding Radius of Convergence ★★☆☆☆ ⏱ 4 min
The standard method for finding $R$ on the AP exam is the Ratio Test, which works seamlessly for factorial, polynomial, and exponential terms common in Taylor and Maclaurin series. For any series $\sum a_n$, the Ratio Test calculates:
L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
The series converges absolutely if $L < 1$, diverges if $L > 1$, and is inconclusive if $L=1$. For a power series, substituting the general term $a_n = c_n (x-a)^n$ gives:
L = |x - a| \cdot \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right|
Requiring $L<1$ for convergence gives the radius of convergence formula:
R = \frac{1}{\lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right|}
The Root Test is an alternative for series with terms raised to the nth power: $R = 1/\left(\lim_{n \to \infty} \sqrt[n]{|c_n|}\right)$, but this is rarely tested on the AP exam. If the limit of the ratio is 0, $R = \infty$ (converges everywhere); if the limit is infinite, $R=0$ (converges only at the center).
Exam tip: Always factor $|x-a|$ out of the limit before evaluating—this term does not depend on $n$, so factoring it out simplifies your limit calculation and avoids algebraic errors.
3. Finding Interval of Convergence: Testing Endpoints ★★★☆☆ ⏱ 5 min
Once you have the radius of convergence $R$, you know the open interval of convergence is $(a-R, a+R)$. For any $x$ inside this open interval, the Ratio Test guarantees absolute convergence. However, at the two endpoints $x = a-R$ and $x = a+R$, $|x-a| = R$, which means $L=1$, and the Ratio Test is inconclusive.
You must test convergence at each endpoint separately using other convergence tests: the nth-Term Test for Divergence, Alternating Series Test, p-Series Test, or Comparison Test. At each endpoint, the series can converge absolutely, converge conditionally, or diverge. You include any endpoint that converges (either absolutely or conditionally) in your final interval. The AP exam explicitly tests whether you remember to check endpoints—omitting this step costs points on FRQ.
Exam tip: When testing endpoints, simplify the series fully before applying a convergence test—$R^n$ almost always cancels out completely, leaving you with a simple alternating or positive series that is easy to test.
4. Edge Cases of Convergence ★★★☆☆ ⏱ 3 min
Two edge cases appear regularly on the AP exam, and both are common sources of lost points. The first edge case is convergence only at the center, which gives $R=0$ and an interval of convergence that is just the single point $\{a\}$. This occurs when coefficients $c_n$ grow so quickly that for any $x \neq a$, the limit $L$ from the Ratio Test is greater than 1, so the series diverges everywhere except the center.
The second edge case is convergence for all real $x$, which gives $R = \infty$ and an interval of convergence of $(-\infty, \infty)$. This occurs when coefficients decay very quickly (most commonly when denominators have factorials, as in the Maclaurin series for $e^x$, $\sin x$, and $\cos x$), so the limit $L$ is 0 for any $x$, which is always less than 1.
A third less common edge case is a power series with non-zero coefficients only for even or odd powers of $(x-a)$; the method for finding $R$ does not change, but you will get an extra factor of $(x-a)^2$ in your ratio, so be careful to simplify correctly.
Exam tip: Always explicitly confirm convergence at the center for $R=0$ cases—remember that every power series converges at its center, even if it diverges everywhere else.
5. Check Your Understanding ★★☆☆☆ ⏱ 2 min
Common Pitfalls
Why: AP exam problems are designed to have one or both endpoints converge, explicitly testing this step, and omitting it loses points
Why: Students drop the absolute value out of habit when working with positive terms, but $(x-a)$ can be negative
Why: Students treat $kx - a$ the same as $x-a$ without adjusting for the coefficient of $x$
Why: Students confuse 'inconclusive' with 'divergent'—the Ratio Test just does not give an answer, not that the answer is divergence
Why: Students are used to working with Maclaurin series (center 0) and forget to adjust for Taylor series centered at a non-zero point