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AP Calculus BC Practice Quiz: The $n^{th}$ Term Test for Divergence

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 7 questions to check your progress.

Question 1 of 7

Consider the series $\sum_{n=1}^{\infty} \frac{3n-1}{n+4}$. What conclusion can be drawn from the $n^{th}$ Term Test for Divergence?

All Questions (7)

Consider the series $\sum_{n=1}^{\infty} \frac{3n-1}{n+4}$. What conclusion can be drawn from the $n^{th}$ Term Test for Divergence?

A) The series converges to 3.

B) The series converges, but its sum cannot be determined by this test.

C) The series diverges.

D) The test is inconclusive.

Correct Answer: C

To apply the $n^{th}$ Term Test, we evaluate the limit of the $n^{th}$ term as $n$ approaches infinity: $\lim_{n \to \infty} \frac{3n-1}{n+4} = 3$. Since the limit is not equal to 0, the $n^{th}$ Term Test concludes that the series diverges.

What does the $n^{th}$ Term Test for Divergence conclude about the series $\sum_{n=1}^{\infty} \frac{1}{n}$?

A) The series converges to 0.

B) The series converges to 1.

C) The series diverges.

D) The test is inconclusive.

Correct Answer: D

First, we find the limit of the $n^{th}$ term: $\lim_{n \to \infty} \frac{1}{n} = 0$. When the limit of the $n^{th}$ term is 0, the $n^{th}$ Term Test for Divergence is inconclusive. It does not provide any information about whether the series converges or diverges. (Note: Although this specific series, the harmonic series, is known to diverge, this conclusion cannot be reached using the $n^{th}$ Term Test.)

The $n^{th}$ Term Test for Divergence is a test used primarily to determine which property of an infinite series?

A) The exact sum of a convergent series.

B) Whether a series converges.

C) Whether a series diverges.

D) The rate at which a series converges.

Correct Answer: C

As its name implies, the $n^{th}$ Term Test for Divergence can only be used to prove that a series diverges. It can never be used to prove that a series converges. If the test is inconclusive (i.e., the limit of the terms is 0), another test must be used to determine convergence or divergence.

For a series $\sum a_n$, it is known that $\lim_{n \to \infty} a_n = L$. Which of the following conditions guarantees that the series diverges by the $n^{th}$ Term Test?

A) L = 0

B) L ≠ 0

C) L > 0

D) L is a finite number

Correct Answer: B

The $n^{th}$ Term Test for Divergence states that if the limit of the sequence of terms, $\lim_{n \to \infty} a_n$, is not zero, the series $\sum a_n$ must diverge. The condition L ≠ 0 covers all cases where the limit is a non-zero number, positive, negative, or infinite.

For which of the following series is the $n^{th}$ Term Test for Divergence sufficient to prove divergence?

A) $\sum_{n=1}^{\infty} \frac{n}{n^2+3}$

B) $\sum_{n=1}^{\infty} \frac{5}{n^p}$, where p > 1

C) $\sum_{n=1}^{\infty} \arctan(n)$

D) $\sum_{n=1}^{\infty} e^{-n}$

Correct Answer: C

We must find the series where the limit of the $n^{th}$ term is not zero. A) $\lim_{n \to \infty} \frac{n}{n^2+3} = 0$. Test is inconclusive. B) $\lim_{n \to \infty} \frac{5}{n^p} = 0$ since p > 1. Test is inconclusive. C) $\lim_{n \to \infty} \arctan(n) = \frac{\pi}{2}$. Since the limit is not 0, the series diverges by the $n^{th}$ Term Test. D) $\lim_{n \to \infty} e^{-n} = \lim_{n \to \infty} \frac{1}{e^n} = 0$. Test is inconclusive.

A student correctly applies the $n^{th}$ Term Test to a series $\sum a_n$ and finds that the test is inconclusive. What must be true about the series?

A) The series converges.

B) The series diverges.

C) $\lim_{n \to \infty} a_n = 0$.

D) $\lim_{n \to \infty} a_n$ does not exist.

Correct Answer: C

The $n^{th}$ Term Test is inconclusive if and only if the limit of the terms of the series is equal to zero. If the limit were non-zero or did not exist, the test would conclude that the series diverges. An inconclusive result means $\lim_{n \to \infty} a_n = 0$, and another test is needed to determine convergence or divergence.

Consider the series $\sum_{n=1}^{\infty} (1 - \frac{1}{n})^n$. What is the conclusion of the $n^{th}$ Term Test for Divergence?

A) The test is inconclusive because the limit of the terms is 0.

B) The series converges because the limit of the terms is $e^{-1}$.

C) The series diverges because the limit of the terms is $e^{-1}$.

D) The test is inconclusive because the limit of the terms is 1.

Correct Answer: C

To apply the $n^{th}$ Term Test, we evaluate the limit of the $n^{th}$ term. We use the known limit form $\lim_{n \to \infty} (1 + \frac{x}{n})^n = e^x$. In this case, $x = -1$. Therefore, $\lim_{n \to \infty} (1 - \frac{1}{n})^n = e^{-1}$. Since $e^{-1} = \frac{1}{e} \neq 0$, the $n^{th}$ Term Test concludes that the series diverges.