Math 172
Exam 3 Review
Apr 2016
1. Find the sum of each of the following series, or show it diverges.
(a)
∞
X
n=2
2
2
n −1
(b)
∞
X
n=2
2
2
n −n
(c)
∞
X
2n+2
n=1
(d)
32n
∞
X
22n
3n+1
n=0
3 9 27 81 243
1
8
82 83 84 85
(f) 1 + + +
+
+
+ ···
+ 2 + 3 + 4 + 4 + 6 + ···
2 4
8
16
32
9 9
9
9
9
9
1
1
1
1
1
1
1
1
1
+ ln − ln
+ ln − ln
+ ln − ln
+ ln − ln
+ ···
(g) ln 1 − ln
2
2
3
3
4
4
5
5
6
(e)
2. Use either comparison test to show the following converge or diverge.
∞
X
1
(a)
2
n −1
n=2
∞ √
X
9n + 3
(b)
n+2
∞
X
3n + 7
√
(c)
3
n5 − 1
n=2
(d)
n=1
∞
X
√
n=1
(e)
π
n + n2
(f)
∞
X
n=1
∞
X
n=1
sin
1
n2
ln n
(n + 1)3
3. Use the integral test to show the following converge or diverge.
(a)
∞
X
n
n2
e
n=1
(b)
∞
X
1
√
3
n=2 n ln n
4. Show the following converge absolutely, converge conditionally, or diverge.
(a)
(b)
∞
X
sin n cos 5n
n=1
∞
X
n=2
n2 + 3
cos(πn)
n−1
(c)
(d)
∞
X
(−1)n−1
n=3
∞
X
n=4
5. Find an error estimate for cos π − 1 −
(e)
n2 − 2n
(−1)n (n − 1)
n+1
π2
2
+
π4
4!
−
π6
6!
(f)
∞ X
2n + 3 n
n=5
∞
X
n=6
8 + π8! <
5n − 7
2n n2
(2n + 5)!
.
6. Find the radius of convergence for the following power series.
(a)
∞
X
nn
n=0
n!
(x − e)n
(b)
∞
X
(2n − 1)!(x + 2)n
(c)
n=1
∞
X
(−1)n x2n
n=0
(2n + 1)!
7. Find the interval of convergence for the following power series.
(a)
(b)
∞
X
xn
n=1
∞
X
n=0
(c)
n
(−1)n
(x + 4)2n
4n
(d)
∞
X
n−1
n=1
∞
X
n2n
(x − 2)n
(e)
n!(x + π)n
(f)
n=0
∞
X
(−1)n x2n
n=0
∞
X
n=1
(2n + 1)!
(6x − 12)n
n2
8. Find the Maclaurin series for the following functions and determine the interval of convergence.
(a) f (x) =
4x
2+x
(b) g(x) =
sin 2x
x
(c) h(x) = xex
2
(d) k(x) =
5
1 + x2
9. Find the first four nonzero terms of the Taylor series of the following centered at c.
(a) f (x) = (5 − x)3/2 ; c = 1
(b) g(x) = sin 2x; c =
π
4
10. Use series to compute the following limits.
x cos x
x→0 1 − ex
sin x − x + x3 /6
x→0
2x5
(b) lim
(a) lim
11. Find the first four nonzero terms of the Maclaurin series of the following.
(a) f (x) = sin 2x + ex
2
(b) g(x) =
ex
1−x
12. Choose all series below that can be shown to diverge using the Divergence Test.
(a)
X (−1)n n
(b)
3−n
X2
n
(c)
X
n4
1
+1
13. True/False.
(a) T / F : If lim an = 7, the series
P
an converges.
(b) T / F : If lim an = 0, the series
P
an converges.
(c) T / F : If lim an = 0, the series
P
an diverges.
n→∞
n→∞
n→∞
P
1
, the series
an diverges.
n
X1
converges.
The Harmonic series
n
X (−1)n+1
converges.
The Alternating Harmonic series
n
P
P
If
|an | converges, then
an converges conditionally.
P
P
If
|an | diverges, then
an diverges.
∞
X (−1)n π 2n+1
=0
(2n + 1)!
n=0
X
X1
1
1 X 1
−
=
−
n n+1
n
n+1
(d) T / F : If 0 < an <
(e) T / F :
(f) T / F :
(g) T / F :
(h) T / F :
(i) T / F :
(j) T / F :
14. For each of the following series determine if they converge or diverge and then choose a test that
can be used to show that.
2
X
1 −n
(a)
1+
Converges / Diverges
by the Root Test / Divergence Test.
n
X
1 n
(b)
1+
Converges / Diverges
by the Root Test / Divergence Test.
n
X 1
(c)
Converges / Diverges
by the Ratio Test / Integral Test.
n ln n
X n+1
(d)
Converges / Diverges
by the Limit Comparison Test / Ratio Test.
n2 + 2
X (−1)n
(e)
Converges / Diverges
by the Integral Test / Alternating Series Test.
n
X
(f)
(−n)n
Converges / Diverges
by the Divergence Test / Alternating Series Test.
15. Use an appropriate Maclaurin series to integrate the following.
Z
Z
sin x
x2
(a)
e dx
(b)
dx
x
Z
(c)
sin x2 dx
16. Choose all series below that can be shown to converge using the Divergence Test.
(a)
X 4 ln n
n2 + 1
(b)
X2
n
(c)
X
1
en + n2
17. Choose all series below that can be shown to converge using either Comparison Test.
(a)
X
1
n ln n
(b)
X (−1)n
n2 + 1
(c)
X ln n
n
18. Choose all series below that can be shown to converge using the Alternating Series Test.
(a)
X (−1)n
√
n
(b)
X cos(πn)
n
(c)
X (−1)n n
n+1
19. Choose all series below that can be shown to diverge using the Alternating Series Test.
X cos n
X (−1)n
X (−1)n n
(a)
(b)
(c)
n
n
n+1
20. Choose all series below that can be shown to converge using the Ratio or Root Test.
X
X
X 2n
n
3n
(a)
(b)
(c)
n3 + 5n
22n+1 + 1
n!
21. Choose all series below that can be shown to diverge using the Ratio or Root Test.
(a)
X
n2
3
n + 5n
(b)
X
3n
2n+1 + 1
(c)
X nn
n!
22. Match a series on the left with a statement on the right to make a true statement.
1−
1 1 1
1
1
1
+ − +
−
+
−···
3 4 9 10 27 28
a) has radius of convergence R = 7.
b) has radius of convergence R = 0.
∞
X
n=1
(−1)n x2n
(2n)!
c) is a convergent alternating series.
d) is a divergent alternating series.
d
dx
∞
X
∞
X
(−1)n+1 x2n
n=0
n=1
e) is the Maclaurin series for cos x − 1.
(2n)!
f) is the Maclaurin series for cos x.
g) is the Maclaurin series for sin x.
133−2n
h) is the Maclaurin series for − sin x.
n=7
∞
X
!
i) is a convergent geometric series.
n
n
n (x − 7)
j) is a divergent geometric series.