AP Calculus BC Practice Quiz: Alternating Series Error Bound
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
All Questions (7)
A) To determine if the series converges or diverges.
B) To calculate the exact sum of the infinite series.
C) To provide an upper limit on the error when a partial sum is used to approximate the infinite sum.
D) To increase the rate of convergence of the series.
Correct Answer: C
The provided content states that the alternating series error bound is used 'to bound how far a partial sum is from the value of the infinite series,' which is the definition of an upper limit or bound on the approximation error.
A) less than or equal to the absolute value of the first term not included in the partial sum.
B) exactly equal to the absolute value of the first term not included in the partial sum.
C) greater than or equal to the absolute value of the nth term.
D) equal to the difference between the nth and (n-1)th terms.
Correct Answer: A
The core principle of the alternating series error bound is that the error of a partial sum approximation is bounded by (i.e., less than or equal to) the magnitude of the first omitted term.
A) The 9th partial sum, `S_9`
B) The magnitude of the 9th term, `a_9`
C) The magnitude of the 10th term, `a_{10}`
D) The magnitude of the 11th term, `a_{11}`
Correct Answer: D
The alternating series error bound states that the error in approximating the sum with the nth partial sum is bounded by the absolute value of the first neglected term. If the 10th partial sum is used, the first neglected term is the 11th term, `a_{11}`.
A) Find the smallest `N` such that the partial sum `S_N` is less than 0.01.
B) Find the smallest `N` such that the magnitude of the Nth term, `|a_N|`, is less than 0.01.
C) Find the smallest `N` such that the magnitude of the next term, `|a_{N+1}|`, is less than 0.01.
D) Find the smallest `N` such that the difference `|S_N - S_{N-1}|` is less than 0.01.
Correct Answer: C
The error for the Nth partial sum, `S_N`, is bounded by the magnitude of the first unused term, `|a_{N+1}|`. To guarantee the error is less than 0.01, one must find the smallest number of terms `N` such that the bound, `|a_{N+1}|`, is less than 0.01.
A) For any series where the terms alternate in sign.
B) Only for a series that has been proven to converge by the alternating series test.
C) For any convergent series, regardless of whether it is alternating.
D) For any divergent alternating series to estimate its partial sums.
Correct Answer: B
The provided content specifies the condition for using the error bound: 'If an alternating series converges by the alternating series test, then the alternating series error bound can be used.' This makes its application conditional on the convergence of the alternating series.
A) An exact calculation
B) A proof of divergence
C) An approximation
D) A test for convergence
Correct Answer: C
The provided content explicitly mentions the goal is to 'Approximate the sum of a series.' A partial sum is a finite sum of terms used to estimate the value of the full infinite sum, making it an approximation.
A) `E_n < E_{n+1}`
B) `E_n > E_{n+1}`
C) `E_n = E_{n+1}`
D) The relationship cannot be determined.
Correct Answer: B
The error bound for `S_n` is `E_n = |a_{n+1}|`, and the error bound for `S_{n+1}` is `E_{n+1} = |a_{n+2}|`. Since the series converges by the alternating series test, the absolute values of the terms are decreasing, meaning `|a_{n+1}| > |a_{n+2}|`. Therefore, the maximum possible error decreases as more terms are added to the partial sum.