Exam III
Calculus II
Name:
Calculators are not allowed on this exam. Work all questions completely. Show all work as described in class.
Copyright 2022 Emilio Verdooren and Texas Tech University. Unauthorized reproduction prohibited.
No credit will be received on answers that are not accompanied with adequate work shown.
1. Determine if the following series converge or diverge. State any tests you use clearly and
show all the work required to use them.
?
X
(a)
32+k 21?3k
(b)
k=0
?
X
2
3 + 5k
k=0
?
X
k2
(c)
k3 ? 3
k=0
(d)
(e)
?
X
2k
k=0
?
X
k!
1
1
?
k k+1
k=1
2. For each of the following sequences determine if (I) the sequence absolutely converges,
(II) the sequence converges but does not converge absolutely, (III) the sequence diverges.
State any tests you use clearly and show all the work required to use them.
?
X
(?1)k+1
(a)
9k + 8
k=1
(b)
?
X
(?1)n+1 (1 + n)2n
k=1
32n n2
3. Find the Maclaurin series of f (x) = 5e4x , show all work. hint: Start taking derivatives
and see if you can come up with a formula for f (n) (x).
P
k
4. Suppose that a power series ?
k=0 ak x converges when x = 2, give two other values of
x for which the power series must converge. Explain why the series will converge for the
values you give (x = 0 doesn count but if you thought of that nice try).
5. Find the first three terms of the Taylor series of f (x) = ln(x3 ) centered at x = 1. Show
all work.
6. Determine if the following statements are true or false (no work necessary, simply state
true or false).
?
X
1
?
(a) If {ak }k=0 converges then
converges
ak
k=0
(b) If limk?? ak = 0 then
?
X
k=0
ak converges.
(c) If {ak }?
k=0 is a positive sequence such that a0 = 1 and
|r| < 1 then
?
X
ak+1
= r for all k ? 0 where
ak
rk converges.
k=0
?
(d) If {ak }?
k=0 and {bk }k=1 are sequences such that 0 ? ak < bk for all k ? 0 and
converges, then
?
X
?
X
bk
k=0
ak converges.
k=0
?
(e) If {ak }?
k=0 and {bk }k=0 are positive sequences such that limk??
and
?
X
k=0
bk both converge or diverge.
?
X
ak
= 0, then
ak
bk
k=0

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