Math 126-104
Test 2
Spring 2013
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; point values are indicated on the problems. Try
adding blueberries to your pancakes.
1. Determine the limit of the sequence (5 points, each):
(a)
an =
sin n
n
(b)
an =
1 − 5n4
n4 − 6n3 + 1
(c)
an =
2n − 1
2n−1
(d)
an =
5n
n!
2. (10 points) Give a formula for the nth partial sum of the series
∞ X
1
k=1
1
−
n n+1
,
and use this formula to sum the series.
3. (5 points) Represent the repeating decimal
1.414414 = 1.414414414414 . . .
as a fraction in lowest terms.
4. Determine the interval of convergence for the power series (10 points each).
(a)
∞
X
(−1)n xn
n=1
n
(b)
∞
X
n=1
(2x)n
5. Use any test that you like to determine if the given series converges (8 points each).
(a)
∞
X
n2
n=2
1
−5
(b)
∞
X
1
en
n=1
(c)
∞ X
4 n
1+
n
n=1
(d)
∞
X
1
n n2 + 4
n=2
√
(e)
∞
X
2n
n=1
n!
6. (10 points) Given that the geometric series
2
1 + x + x + ··· =
∞
X
xn =
n=0
1
,
1−x
use substitution (x = −y, y = z 2 ) and term-by-term integration to obtain a series for
f (x) = arctan (x).