MATH 172.503
Examination 3
April 22, 2010
NAME
SIGNATURE
This exam consists of 10 problems, numbered 1–10. For partial credit you must present
your work clearly and understandably and justify your answers.
The use of calculators is not permitted on this exam.
The point value for each question is shown next to each question.
CHECK THIS EXAMINATION BOOKLET BEFORE
YOU START. THERE SHOULD BE 10 PROBLEMS
ON 7 PAGES (INCLUDING THIS ONE).
Do not mark in the box below.
1–5
6
7
8
9
10
Total
Points
Possible Credit
30
12
10
10
18
20
100
NAME
MATH 172.503
Examination 3
Page 2
Multiple Choice: [6 points each] In each of Problems 1–5, circle the best answer.
1.
Find the formula for the n-th term of the sequence
1 1 1
1 1
,− , ,− , ,....
2 5 10 17 26
(A)
(−1)n+1
3n − 1
(B)
(−1)n+1
2 + 3n−1
(C)
(−1)n
(n + 1)!
(D)
(−1)n+1
n2 + 1
n
2
1
(E) · −
2
5
2.
Evaluate lim
n→∞
cos2 (n)
3
n
1
+
.
3
(A) 0
(B)
1
3
(C) 1
(D)
2
3
(E) The sequence diverges.
April 22, 2010
NAME
3.
MATH 172.503
The series
Examination 3
Page 3
∞
X
22n − 1
n=0
5n+1
(A) converges to 0.
(B) converges to
5
.
3
(C) converges to
3
.
4
(D) converges to
3
.
5
(E) diverges.
4.
Define the sequence {an } by an =
values of N :
4−n
, and let L = lim an . Consider the following
n→∞
n
I. N = 100,
II. N = 400,
III. N = 1000.
In the definition of a limit we decide to pick =
so that the following statement is true?
“For every n > N , we have |an − L| <
1
.
100
What value(s) of N can we pick
1
.”
100
(A) I, II, and III
(B) II and III only
(C) III only
(D) I and II only
(E) none of I, II, or III
April 22, 2010
NAME
5.
MATH 172.503
Consider the series
Examination 3
Page 4
∞
X
(−1)n+1
. By the Alternating Series Estimation Theorem,
n2
what is an upper bound on the error obtained when estimating the total sum of this
series by the sum of its first 10 terms?
n=1
(A) at most
1
121
(B) at most
1
100
(C) at most
1
81
(D) at most
1
10
(E) the series diverges
6.
[12 points] Consider the series
∞ X
1
n=1
1
−
n n+2
.
(a) Find a formula for the partial sum sk of the series.
(b) Find the sum of the series or show that the series diverges.
April 22, 2010
NAME
7.
MATH 172.503
Examination 3
Page 5
[10 points] Find the limits of the following sequences.
(a) an =
3n
n3
(b) bn = ln(3n) − ln(n − 2)
8.
[10 points] Find a power series representation for the function
f (x) =
x
,
1 − x2
which is valid for |x| < 1.
April 22, 2010
NAME
9.
MATH 172.503
Examination 3
Page 6
[18 points] Determine whether each series below converges or diverges. For each
problem use one or more of the following tests: ratio test, test for divergence (n-th
term test), direct comparison test, limit comparison test, integral test, p-series test.
(Be sure to specify which test(s) you are using and show how they apply.)
(a)
∞
X
tan−1 (n)
3n
n=1
∞
X
1
cos
(b)
n
n=1
(c)
∞
X
n=2
√
n4 − 1
n5
April 22, 2010
NAME
10.
MATH 172.503
Examination 3
Page 7
[20 points] Consider the following function:
f (x) =
∞
X
(x − 2)n
n=1
n · 2n
.
(a) What is the open interval of convergence for f (x)?
(b) Does the series above converge at the endpoints of the interval in (a)?
(c) Write down a power series expression for f 0 (x).
(d) Find f 0 (3).
April 22, 2010