Math 231
Practice Midterm #6
Fall 2020
On PrairieLearn, the questions below may appear either as free response or multiple choice.
These questions should give you a general idea of the format of the exam. You should also study
any questions that have appeared in worksheets or WebAssign HW for this material.
1. Error Estimates (2 - 3 questions will be chosen of this type)
Version A: Which of the following partial sums contains the fewest number of terms needed
∞
X
(−1)n
to approximate
with an error of less than 1/1000 ?
10n3
n=1
(A)
4
X
(−1)n
n=1
(B)
6
X
(−1)n
n=1
(C)
10n3
2
X
(−1)n
n=1
(E)
10n3
8
X
(−1)n
n=1
(D)
10n3
10n3
10
X
(−1)n
n=1
10n3
Version B: What is the smallest value of N such that the partial sum SN approximates the
∞
X
(−1)n
sum S =
to within 1/50?
n2
n=1
Version C: What is the maximum error when the 30th partial sum S30 is used to estimate
∞
X
1
? Type your answer as a decimal rounding to 4 decimal places.
n3
n=1
Version D: Find the value of N which guarantess the N th partial sum of the series approxi1
mates the sum with an error ¡ 96
.
2. Convergence, Conditional Convergence, Divergence (2 - 3 questions will be chosen of this type)
Version A: Consider the series
∞
X
(−1)n 5n
n!
n=0
. Which of the following is true?
(A) The series converges absolutely by the ratio test.
(B) The series converges by the alternating series test, but does not converge absolutely.
(C) The series diverges by the test for divergence.
(D) The series converges absolutely by the alternating series test.
(E) The series diverges by the ratio test.
Version B: Consider the series
∞
X
3n + 1
. Which of the following is true?
(−1)n+1
8n
+
2
n=0
(A) The series diverges by the test for divergence.
(B) The series diverges by the ratio test.
(C) The series converges by the alternating series test, but does not converge absolutely.
(D) The series converges absolutely by the alternating series test.
(E) The series converges conditionally by the ratio test.
Version C: Consider the series
∞
X
(−1)n
n=1
n2/3
. Which of the following is true?
(A) The series converges conditionally.
(B) The series converges absolutely.
(C) The series diverges.
Math 231
Practice Exam #5
Page 2 of 3
∞
X
(n!)2
Version D: Consider the series
. Which of the following is true?
(2n)!
n=1
(A) The series converges conditionally.
(B) The series converges absolutely.
(C) The series diverges.
Math 231
Practice Exam #5
Page 3 of 3