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AP Calculus BC Practice Quiz: Defining Convergent and Divergent Infinite Series

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

Test your understanding with short quizzes. This quiz has 9 questions to check your progress.

Question 1 of 9

For an infinite series $\sum_{k=1}^{\infty} a_k$, what is the definition of the nth partial sum, denoted by $S_n$?

All Questions (9)

For an infinite series $\sum_{k=1}^{\infty} a_k$, what is the definition of the nth partial sum, denoted by $S_n$?

A) The value of the nth term, $a_n$.

B) The sum of the first $n$ terms, $a_1 + a_2 + ... + a_n$.

C) The limit of the sequence of terms as $k$ approaches $n$.

D) The sum of the entire infinite series.

Correct Answer: B

Based on the provided content, 'The nth partial sum is defined as the sum of the first $n$ terms of a series.'

An infinite series $\sum_{n=1}^{\infty} a_n$ is said to converge to a sum $S$ if which of the following conditions is met?

A) The sequence of terms \{a_n\} converges to $S$.

B) The sequence of terms \{a_n\} converges to 0.

C) The sequence of partial sums \{S_n\} converges to $S$.

D) The sequence of partial sums \{S_n\} is always increasing.

Correct Answer: C

The provided content states, 'An infinite series of numbers converges to a real number $S$... if and only if the limit of its sequence of partial sums exists and equals $S$.'

Let \{$S_n$\} be the sequence of partial sums for the infinite series $\sum_{k=1}^{\infty} a_k$. If $\lim_{n \to \infty} S_n = 5$, what can be concluded about the series?

A) The series converges to 5.

B) The series diverges.

C) The series converges, but its sum cannot be determined from the given information.

D) The nth term of the series, $a_n$, is equal to 5.

Correct Answer: A

By definition, an infinite series converges to a sum $S$ if the limit of its sequence of partial sums equals $S$. Since the limit of the partial sums is 5, the series converges to 5.

The nth partial sum of an infinite series is given by $S_n = \frac{2n}{n+1}$. Does the series converge or diverge, and if it converges, what is its sum?

A) The series converges to 0.

B) The series converges to 1.

C) The series converges to 2.

D) The series diverges.

Correct Answer: C

The sum of the series is the limit of its sequence of partial sums as $n \to \infty$. We need to calculate $\lim_{n \to \infty} S_n = \lim_{n \to \infty} \frac{2n}{n+1}$. By comparing the degrees of the numerator and denominator, this limit is the ratio of the leading coefficients, which is $2/1 = 2$. Therefore, the series converges to 2.

Let \{$S_n$\} be the sequence of partial sums for an infinite series. Which of the following conditions implies that the series diverges?

A) $\lim_{n \to \infty} S_n = 0$.

B) The sequence \{$S_n$\} is bounded.

C) The sequence \{$S_n$\} is decreasing.

D) The limit of the sequence \{$S_n$\} does not exist.

Correct Answer: D

A series converges if and only if the limit of its sequence of partial sums exists and is a finite real number. If this limit does not exist (either because it approaches infinity or oscillates), the series diverges.

Which of the following statements accurately describes the relationship between an infinite series and its sequence of partial sums?

A) A series converges if its corresponding sequence of partial sums converges.

B) A series converges if its corresponding sequence of terms, \{$a_n$\}, converges.

C) The sum of a series is equal to the last term of its sequence of partial sums.

D) The nth partial sum is the limit of the first $n$ terms of the series.

Correct Answer: A

This is the fundamental definition provided in the content. An infinite series converges to a sum $S$ if and only if its sequence of partial sums, \{$S_n$\}, converges to $S$. The other options are incorrect interpretations.

The nth partial sum of an infinite series is given by $S_n = 3 - \frac{1}{n^2}$. Based on this information, which of the following is true?

A) The series converges to 0.

B) The series converges to 3.

C) The series diverges because the partial sums are always less than 3.

D) The series diverges because $\lim_{n \to \infty} \frac{1}{n^2}$ does not exist.

Correct Answer: B

To find the sum of the series, we must evaluate the limit of the sequence of partial sums. $\lim_{n \to \infty} S_n = \lim_{n \to \infty} (3 - \frac{1}{n^2})$. As $n \to \infty$, the term $\frac{1}{n^2}$ approaches 0. Therefore, the limit is $3 - 0 = 3$. The series converges to 3.

The first four partial sums of the series $\sum_{k=1}^{\infty} a_k$ are $S_1 = 2$, $S_2 = 5$, $S_3 = 4$, and $S_4 = 10$. What is the value of the third term, $a_3$?

A) -1

B) 1

C) 4

D) 9

Correct Answer: A

The nth partial sum is the sum of the first $n$ terms: $S_n = a_1 + a_2 + ... + a_n$. Therefore, $S_3 = a_1 + a_2 + a_3$ and $S_2 = a_1 + a_2$. The third term, $a_3$, can be found by calculating $S_3 - S_2$. Using the given values, $a_3 = 4 - 5 = -1$.

If an infinite series is known to converge, what must be true about its sequence of partial sums, \{$S_n$\}?

A) The sequence \{$S_n$\} must approach infinity.

B) The sequence \{$S_n$\} must approach a finite limit.

C) The sequence \{$S_n$\} must be strictly increasing.

D) The sequence \{$S_n$\} must oscillate between two values.

Correct Answer: B

The definition of a convergent series states that it converges to a real number $S$ if and only if the limit of its sequence of partial sums exists and equals $S$. This means the sequence of partial sums must approach a finite limit.