Math 126-104 (CRN 31611)
Test 2
July 13, 2015
Carter
General Instructions: Write your name on only the outside of your blue book. Do all of your
work and write your answers inside your blue book. Put your test paper inside your blue book as
you leave. Solve each of the following problems. There are 115points. Make sure your solutions
are written neatly, and I can recognize the answer. A simple wedge of lettuce with a side dish of
dressing is a refreshing salad in the summer.
1. Last week’s meme on the social media that I frequent was a photo of a bumper sticker that
read, “If you can compute
Z ∞
dx
,
2
−∞ x + 1
then you are following too closely.” (8 points) Compute this improper integral.
2. (8 points) Compute
Z
1
dx .
(x + 1)2 (x − 1)
3. Determine the limit of the sequence (5 points each):
(a)
an =
ln n
n
(b)
an =
n2 − 2
3n2 + 7n + 6
(c)
an = 2 + (0.1)n
(d)
an =
2n
n!
4. (10 points) Give a formula for the nth partial sum of the series
∞ X
1
1
−
,
k (k + 1)
k=1
and use this formula to sum the series.
5. (5 points) Represent the repeating decimal
0.24 = 0.2424242424 . . .
as a fraction in lowest terms.
6. Determine the interval of convergence for the power series (10 points each).
(a)
∞
X
(−1)n x2n+1
n=0
(2n + 1)!
(b)
∞
X
n xn−1
n=1
7. Use any test that you like to determine if the given series converges (8 points each).
(a)
∞
X
n=0
n2
1
+1
(b)
∞
X
(n2 )
n=1
n!
(c)
∞
X
1
n1.2
n=1
8. (10 points) Given that the geometric series
2
1 + y + y + ··· =
∞
X
yn =
n=0
1
,
1−y
use substitution (y = −z, z = x2 ) and term-by-term integration to obtain a series for
f (x) = arctan (x).
9. (10 points) Use the series for f (x) = ex to find a series for g(x) = x3 ex .