Math 1B, Fall 2008, Wilkening
Midterm 2, version 1,3,5
0. (1 point) write your name, your GSI’s name, and your section number at the top
of your exam.
P
2n
1. (3 points or 0 points) Suppose |cos x| =
6 1. Evaluate ∞
n=0 (cos x) .
a. cot x
b. csc2 x
c. cosh x
x
d. √1−x
2
e. none of the above
2. (3 points or 0 points) Describe the behavior of the sequence a1 = 0, an+1 =
a.
b.
c.
d.
e.
a2n + 3
4
an increases monotonically and converges to 1
an increases monotonically and converges to 3
an increases monotonically to ∞
an decreases monotonically to −∞
an is not monotonic
3. (3 points or 0 points) Supppose 0 < an < 1, an < bn , and
Circle all the statements that are necessarily true:
P 2
P 2 P
a.
bn converges and
b < bn
P√
Pn √
P
bn converges and
bn < bn
b.
P 2
P 2 P
c.
an converges and
a < an
P√
Pn√
P
d.
an converges and
an < an
P
e. if p > 0 then (−1)n apn is convergent
2
P
bn is convergent.
4a. (3 points) Let f (x) = e−x .
Write down the Maclaurin series for f (x) and evaluate f (99) (0) and f (100) (0).
4b. (4 points) Find all x that satisfy the equation
∞
X
1
nxn = .
2
n=1
5. (2 points each) For each of the following series, determine whether the series is
absolutely convergent (AC), conditionally convergent (CC), or divergent (D). Show
some work, but do not spend excessive time justifying all your steps.
∞
X
(−1)n [sin(1/n2 )]2/3
n=1
∞
X
ln cos
n=1
1
n
∞
X
(2n)!
3n (n!)2
n=1
∞ X
5
n=0
n
3n
6a. (2 pts) Is the following statement True or False? Justify your answer with a proof
or counterexample. (Obviously it’s true if an ≥ 0 and bn ≥ 0, so don’t assume this).
P
P
an is divergent and
bn is divergent, then (an + bn ) is also divergent.
P
P
n
6b. (3 points) Suppose
cn xn has radius of convergence 2 while
dP
n x has radius
of convergence 5. What is the radius of convergence of the series (cn + dn )xn ?
Explain.
If
P
7a. (3 points) Prove that e ≥
1
1+
k
k
for k ≥ 1. Hint: ln(1 + x) = x −
x2
2
+···
7b. (3 points) Use part (a) and mathematical induction to prove the following crude
version of Stirling’s approximation:
en n! ≥ nn
for all n ≥ 1.
Math 1B, Fall 2008, Wilkening
Midterm 2, version 2,4,6
0. (1 point) write your name, your GSI’s name, and your section number at the top
of your exam.
P
2n
1. (3 points or 0 points) Suppose |sin x| =
6 1. Evaluate ∞
n=0 (sin x) .
a. tan x
b. sec2 x
c. sinh x
x
d. √1−x
2
e. none of the above
2. (3 points or 0 points) Describe the behavior of the sequence a1 = 2, an+1 =
a.
b.
c.
d.
e.
a2n + 3
4
an increases monotonically and converges to 3
an decreases monotonically and converges to 1
an increases monotonically to ∞
an decreases monotonically to −∞
an is not monotonic
3. (3 points or 0 points) Supppose 0 < an < 1, an < bn , and
Circle all the statements that are necessarily true:
P 2
P 2 P
a.
an converges and
a < an
P√
Pn√
P
an converges and
an < an
b.
P 2
P 2 P
c.
bn converges and
b < bn
P√
Pn √
P
d.
bn converges and
bn < bn
P
e. if p > 0 then (−1)n apn is convergent
3
P
bn is convergent.
4a. (3 points) Let f (x) = e−x .
Write down the Maclaurin series for f (x) and evaluate f (99) (0) and f (100) (0).
4b. (4 points) Find all x that satisfy the equation
∞
X
n=1
nxn = 1.
5. (2 points each) For each of the following series, determine whether the series is
absolutely convergent (AC), conditionally convergent (CC), or divergent (D). Show
some work, but do not spend excessive time justifying all your steps.
∞
X
[sin(1/n2 )]1/3
n=1
∞
X
(−1)n ln cos
n=1
1
n
∞
X
(2n)!
5n (n!)2
n=1
∞ X
5
n=0
n
(−3)n
6a. (2 pts) Is the following statement True or False? Justify your answer with a proof
or counterexample. (Obviously it’s true if an ≥ 0 and bn ≥ 0, so don’t assume this).
P
P
an is divergent and
bn is divergent, then (an + bn ) is also divergent.
P
P
n
6b. (3 points) Suppose
cn xn has radius of convergence 2 while
dP
n x has radius
of convergence 1. What is the radius of convergence of the series (cn + dn )xn ?
Explain.
If
P
7a. (3 points) Prove that e ≥
1
1+
k
k
for k ≥ 1. Hint: ln(1 + x) = x −
x2
2
+···
7b. (3 points) Use part (a) and mathematical induction to prove the following crude
version of Stirling’s approximation:
en n! ≥ nn
for all n ≥ 1.
results.txt
Tue Nov 04 23:56:56 2008
1
results of second midterm:
# students:
345
total possible:
max score:
average:
standard dev:
36 points
32 points
14.09
5.93
rough grading scale (worry about +/- later)
raw score
19-32
14-18
10-13
6-9
1-5
grade
A
B
C
D
F
# of students
88
88
81
71
17
curve for comparison to other midterm and final:
curved score = 35 + 2.75 * raw score
histogram:
raw score
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
# students
1
0
5
5
6
19
18
16
18
14
21
27
19
21
17
15
18
17
18
21
13
5
9
9
5
1
1
2
2
0
1
1
running total
1
#
1
6
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11
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17
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337
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338
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341
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345
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the answers below were written up by Peter Mannisto and James Tener.