Math 1220 (Underdown)
Exam 3
Name:
1. [10 points] Find the limit:
2. [10 points] Find the limit:
limπ
x→ 2
cos x
x − π2
t ln t
t→∞ 1 + t2
lim
April 15, 2015
Math 1220
Exam 3, Page 2 of 8
April 15, 2015
3. [10 points] Find the following limit.
Hint: This is an indeterminate ∞0 form, so let y = (tan x)cos x and find
lim − (tan x)cos x
x→π/2
lim ln y.
x→π/2−
Math 1220
Exam 3, Page 3 of 8
4. [10 points] Evaluate the improper integral or show that it diverges.
Hint: u–substitution.
Z∞
ln x
dx
x
e
5. [10 points] Evaluate the improper integral or show that it diverges.
Z2
0
dx
√
x
April 15, 2015
Math 1220
Exam 3, Page 4 of 8
April 15, 2015
6. [5 points] Determine the sum of the following geometric series.
∞ n−1
X
4
n=1
5
7. [5 points] Determine whether the following series diverges or converges. Explicitly state
your reasoning. If it converges, find its sum.
∞
X
2n
n=1
3n
Math 1220
Exam 3, Page 5 of 8
April 15, 2015
8. [10 points] Use the Limit Comparison Test to determine the convergence or divergence
of the series
∞
X
n2 + 1
22 + 1 32 + 1 42 + 1
=
1
+
+ 3
+ 3
+ ···
3+1
3+1
n
2
3
+
1
4
+
1
n=0
Hint: When n is large adding 1 is insignificant, and each term behaves almost exactly
like a function without the 1s, that is like n2 /n3 . Use this fact to help you choose the
comparison series.
Math 1220
Exam 3, Page 6 of 8
April 15, 2015
9. [10 points] Use the Ratio Test to determine whether the following series converges or
diverges. Explicitly state your reasoning.
∞
X
8n
n=1
n!
Math 1220
Exam 3, Page 7 of 8
April 15, 2015
10. [10 points] Classify the following series as either: divergent, conditionally convergent or
absolutely convergent. Justify your answer by clearly indicating which test(s) you use
and showing that all hypotheses are met.
Check one: divergent
conditionally convergent
∞
X
(−1)n
3n2 + 5
n=1
absolutely convergent
Math 1220
Exam 3, Page 8 of 8
Question Points Score
1
10
2
10
3
10
4
10
5
10
6
5
7
5
8
10
9
10
10
10
Total:
90
April 15, 2015