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Math 21, Fall 2022 Midterm Exam I Practice I Page 1 of 7
1. (15 points) Use the blank lines at the end of each statement to indicate whether the statement is
always true (T) or not always true (F). Only your answer will be graded.
∞
P ∞
P ∞
P
(a) (5 points) Let {an } and {bn } be sequences. If an = 3 and bn = 4, then an bn = 12.
n=1 n=1 n=1
Answer:
(b) (5 points) Let {an } be a sequence. If lim an = 5, then lim n1 ln(1 + an ) = 0.
n→∞ n→∞
Answer:
R∞
(c) (5 points) If twoR continuous functions f and g satisfy f (x) ≤ g(x) for any x ≥ 0 and 0 g(x)dx
∞
converges, then 0 f (x)dx converges. Answer:
2. (5 points) Let {an }, {bn }, and {cn } be three sequences and assume the following:
lim an = 1, lim bn = 2, lim cn = 3.
n→∞ n→∞ n→∞
Decide which of the following choices correctly describes the value of lim (an + 1)(b2n + cn ). Write
n→∞
your choice into the blank at the end of this prompt. Only your answer will be graded. Answer:
(A) 10 (B) 14 (C) 26 (D) Cannot be determined
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Math 21, Fall 2022 Midterm Exam I Practice I Page 2 of 7
3. (15 points) Evaluate the following integrals. Be sure to fully justify your solutions.
Z +∞
dx
(a) (7 points)
e x(ln x)2
Z 3
dx
(b) (8 points) √
−1 x+1
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Math 21, Fall 2022 Midterm Exam I Practice I Page 3 of 7
4. (15 points) Use the Comparison Test to show whether the following integrals converge or diverge.
Z +∞
x+2
(a) (8 points) dx
5 x2 − 16
+∞
x−2
Z
(b) (7 point) dx
5 x3 + x
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Math 21, Fall 2022 Midterm Exam I Practice I Page 4 of 7
5. (15 points) Find explicit formulas for the following sequences:
(a) (5 points) a1 = 0, an = an−1 + 7 for n > 1
an−1
(b) (5 points) a1 = 5, an = for n > 1
4
(c) (5 points) {1, 3, 1, 3, 1, 3, . . . }
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Math 21, Fall 2022 Midterm Exam I Practice I Page 5 of 7
6. (15 points) Let {an } be a sequence and write your answer into the blank at the end of each line. Fully
justify your answers.
2n + en
(a) (5 points) If an = , then lim an = .
n − en n→∞
(b) (5 points) If an = ln(n) · e−n , then lim an = .
n→∞
n+1
(c) (5 points) If an = e n , then lim an = .
n→∞
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Math 21, Fall 2022 Midterm Exam I Practice I Page 6 of 7
√
7. (10 points) Let a1 = 2 and an = 2an−1 − 1. Given that this sequence converges, find lim an .
n→∞
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Math 21, Fall 2022 Midterm Exam I Practice I Page 7 of 7
∞
P n2
8. (10 points) If the n-th partial sum of a series an is given by Sn = 3 − 2n , find an and the sum of
n=1
the series.
