Math 1220-1
Name:
Practice Exam 2
ST#
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1. Find the following:
4x
tan x
(a) lim
x−→0
(b)
(c)
lim t1/t
t−→∞
Z
1
Z
4
e2x dx
−∞
(d)
1
dx
√
x−1
2. (10 pts) Find the following limits:
(a) lim
n→0
2n − sin n
.
n
n100
.
n→∞ en
(b) lim
3. (10 pts) Evaluate the improper integral
Z
1
∞
2
2x e−x dx.
4. (5 pts) Recall the definition for the Fibonacci Sequence:
f1 = 1,
f2 = 1,
fn = fn−2 + fn−1 .
Write out the Fibonacci Sequence through the tenth term.
5. Show that the series
∞
X
n=1
n
(−1)
2n
converges absolutely.
n!
6. (10 pts) Recall the geometric series:
1 + r + r2 + r3 + r4 + · · · =
1
,
1−r
(a) Use this to find the series representation for f (x) =
(b) What is the radius of convergence for this series?
1
.
1+x
|r| < 1.
7. Using the series for
1
1
.
from problem 6, find a series representation for
1+x
(1 + x)2
8. Determine whether the given series converges or diverges, give your reasons, and if it converges, find
its sum if possible.
∞ X
1
1
−
(a)
k k+2
k=1
(b)
j
∞ X
1
ln 2
j=1
(c)
∞
X
i+5
1
+ i3
i=1
NOTE: Know how to use all of the convergence and divergence tests.