# Math 126-104 Test 2 Spring 2013 Carter

```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
you leave. Solve each of the following problems; point values are indicated on the problems. Try
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 + &middot;&middot;&middot; =
∞
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).
```