Calculus BC · Infinite Sequences and Series · 14 min read · Updated 2026-05-11
The nth term test for divergence — AP Calculus BC
AP Calculus BC · Infinite Sequences and Series · 14 min read
1. Definition and Core Purpose★☆☆☆☆⏱ 3 min
The nth term test for divergence (often shortened to the nth term test) is the most fundamental first-step convergence test for infinite series, regularly tested in both multiple-choice and free-response sections of the AP Calculus BC exam, accounting for 1-3% of total exam weight. It leverages a core necessary condition for convergence to quickly eliminate obviously divergent series before you apply more complex, computationally intensive tests.
Exam tip: Always apply the nth term test first when checking any series for convergence. A 10-second limit calculation can save you 5 minutes of unnecessary computation on a clearly divergent series.
2. Core Theorem and Underlying Logic★★☆☆☆⏱ 4 min
To understand why the nth term test works, we start with the definition of infinite series convergence: a series $\sum_{n=1}^{\infty} a_n$ converges if and only if its sequence of partial sums $S_k = \sum_{n=1}^k a_n$ converges to a finite limit $L$ as $k \to \infty$. By definition of partial sums, the $k$th general term can be written as:
a_k = S_k - S_{k-1}
If $\lim_{k \to \infty} S_k = L$ and $\lim_{k \to \infty} S_{k-1} = L$, we can take the limit of both sides:
\lim_{k \to \infty} a_k = \lim_{k \to \infty} (S_k - S_{k-1}) = L - L = 0
This proves that for any convergent series, the limit of the $n$th term must be zero. This is a *necessary condition* for convergence—you cannot have a convergent series without it. The nth term test is just the logically equivalent contrapositive of this statement: if the limit of the $n$th term is not zero, the series cannot be convergent, so it must diverge. This logical equivalence means the test is always valid when applied correctly.
Exam tip: Always evaluate the limit of the nth term before moving to any other convergence test.
3. Interpreting Inconclusive Test Results★★☆☆☆⏱ 4 min
The most commonly tested aspect of the nth term test is understanding what it means when $\lim_{n \to \infty} a_n = 0$. In this case, the test is *inconclusive*—it cannot tell you whether the series converges or diverges. This is because the condition that $\lim_{n \to \infty} a_n =0$ is necessary, but not sufficient, for convergence. In other words, all convergent series have $\lim_{n \to \infty} a_n =0$, but not all series with $\lim_{n \to \infty} a_n =0$ converge. A famous example is the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$: its general term approaches 0, but the series diverges. By contrast, the p-series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ also has a general term that approaches 0, but it converges. When the limit is 0, you must always use a different test (like the integral test or comparison test) to resolve convergence.
Exam tip: On FRQ justifications, never write 'the series converges because $\lim_{n \to \infty} a_n = 0$'—this will always lose points.
4. Non-Existent Limits and Divergence★★★☆☆⏱ 3 min
Many students forget that the nth term test applies not just when the limit exists and is non-zero, but also when the limit of $a_n$ as $n \to \infty$ does not exist (DNE). Common cases are oscillating sequences that do not approach a single fixed value, and unbounded sequences that grow without bound. In both cases, the condition $\lim_{n \to \infty} a_n = 0$ fails, so the necessary condition for convergence is not met, and the series diverges. This is a straightforward application that is often missed by students who only look for finite non-zero limits.
Exam tip: Always check if the limit exists before concluding anything. Oscillating or unbounded general terms always result in divergence, even if terms equal zero infinitely often.
5. AP-Style Worked Practice Problems★★★☆☆⏱ 4 min
Common Pitfalls
Why: Students confuse the necessary condition for convergence with a sufficient condition, since the nth term test is the first convergence test they learn.
Why: Students mix up the definition of series vs partial sums, especially on exam questions about partial sum behavior.
Why: Students confuse small numerical values of $a_n$ for large finite $n$ with a limit of zero as $n \to \infty$.
Why: Students mix up the nth term test with the ratio test early in the unit.
Why: Students see occasional zero terms so they incorrectly assume the limit is zero.