Review for Exam 4 / Chapter 9
Calculus II (MATH 166)
1. Find the limits as n → ∞ of the sequences defined below, or state (and justify) that
it does not exist.
cosn (πn)
,n≥1
(a) an =
n
n
3
(b) bn = 1 + 2 , n ≥ 1
n
(c) cn+1 =
(d) dn =
√
1 + cn for all n ≥ 1, c1 = 2
1
1−
4
1
1
1
1−
1−
··· 1 − 2 , n ≥ 2
9
16
n
2. Determine whether the following series converge or diverge.
(a)
∞
X
arctan(n)
n=1
(b)
1 + n2
∞
X
sin(n3 + 4n + 1)
n3
n=1
(c)
∞
X
nn
n=1
n!
(
nn
(d)
an , where an =
0
n=1
∞
X
if n ∈ {1, 2, ..., 100,000,000}
if n ∈ {100,000,001, ...}
1
3. Find the intervals of convergence for the following power series. Make sure that you
check the end points of the intervals.
∞
X
(−1)n 2n+1
x
(a)
(2n
+
1)!
n=0
∞
X
(2x + 4)n
√
(b)
n n2 + 1
2
n=0
(c)
∞
X
n=2
1 n
x
ln(n)
4. (a) Moe, Curly, and Larry are practicing basketball when they decide to start a game.
In the given order (Moe, Curly, Larry), they will each take a turn at attempting a
free-throw shot. They will keep cycling through the order until one of them makes
the shot and wins the game. Moe can make a free-throw shot with probability 14 ,
Curly with probability 35 , and Larry with probability 21 . Who is the most likely
person to win this game?
(b) Shemp now replaces Curly in this game (the order is now Moe, Shemp, Larry).
Moe and Larry have the same probabilities of making a free-throw shot as in the
previous game. It turns out that for this game, all three players are equally likely
to win. What is the probability that Shemp will make any given free-throw shot?
5. Prove, using mathematical induction, that
[ln(n)]p
=0
n→∞
n
lim
for every p ∈ {1, 2, 3, ...}. That is, show that:
ln(n)
(Base Case) limn→∞
= 0, and
n
(Inductive Step) If we assume that limn→∞
limn→∞
[ln(n)]p+1
= 0.
n
2
[ln(n)]p
= 0 for some p ≥ 1, then
n
6. Use the Ordinary Comparison Test along with Problem 5 to show that
∞
X
n=2
1
[ln(n)]p
diverges for every p ∈ {1, 2, 3, ...}.
7. The imaginary unit i is the complex number defined by i2 = −1. Note that i3 = −i
and ij+4n = ij for each j ∈ {0, 1, 2, 3} and each natural number n. Prove Euler’s
formula. That is, show that for each x,
eix = cos(x) + i sin(x),
and in particular, eiπ = −1.
Z
x
2
8. Determine the Maclaurin series of the function f (x) =
e−t dt (You may use the
−∞
√
R 0 −t2
π
fact that −∞ e dt = 2 ). Use the Maclaurin polynomial of order 3 to approximate
f (0.2).
9. For the function f (x) = ln(x), let Rn (x) = f (x) − Pn (x) be the error of Pn (x), the
nth-order Taylor polynomial of f (x) based at a = 1.
(a) Find the maximum of |f (n+1) (x)| on [1, 23 ] for each n.
(b) Show that |Rn (x)| ≤
|x − 1|n+1
for all x > 1.
n+1
(c) Use the previous parts to approximate ln( 23 ) with an error less than 0.01.
3
Solutions
1. (a) 0
(b) 1
(c)
(d)
√
1+ 5
2
1
2
2. (a) Converges
(b) Converges
(c) Diverges
(d) Converges
3. (a) Converges everywhere
(b) −3 ≤ x < −1
(c) −1 ≤ x < 1
4. (a) Curly
(b)
1
3
5. Hint: Use L’Hôpital’s Rule in both parts.
6. Hint: How does Problem 5 show you that
number N ?
1
n
≤
1
[ln(n)]p
for all n ≥ N for some natural
7. Hint: Use the power series of ex , sin(x), and cos(x).
√
∞
√
3
π X (−1)n x2n+1
+
, f (0.2) ≈ 2π + 0.2 − (0.2)
8. f (x) =
≈ 1.08356
3
2
(2n
+
1)n!
n=0
9. (a) |f (n+1) (x)| ≤ n!
(b) Hint: Use an upper bound for the remainder formula.
(c) ln( 32 ) ≈
1
2
−
( 12 )2
2
+
( 12 )3
3
−
( 12 )4
4
≈ 0.401
4