NAMES:
MATH 152
April 15, 2015
QUIZ 9
• Show all your work and indicate your final answer clearly. You will be graded not merely
on the final answer, but also on the work leading up to it.
1. (3 points) Evaluate the indefinite integral as a power series.
Z
t
dt
1 − t9
Solution:First we find a power series representation of the integrand:
∞
∞
X
X
1
t
9n
t
=
t9n+1
=
t
·
=
t
1 − t9
1 − t9
n=0
n=0
and integrating the sum gives
∞
X
t9n+2
9n + 2
n=0
2. (3 points) Find a power series representation for the function
x
f (x) = 2
7x + 1
Solution:
x
x
1
= 2·
−1
2
7x + 1
7x 1 − 7x
2
n
∞ X
7
1
=
− 2
x n=0
7x
∞
7 X (−1)n
=
x n=0 7n x2n
=
∞
X
n=0
(−1)n
7n−1 x2n+1
NAMES:
MATH 152
3. (3 points) Find the radius of convergence and the interval of convergence of
∞
X
xn
8n − 1
n=1
Solution: We use the Ratio Test and want the limit less than one:
n+1
x (8(n + 1) − 1) an+1 = lim lim n→∞ an n→∞ xn (8n − 1)
x(8n + 7) = lim n→∞
8n − 1 = |x| < 1
so convergence happens when |x| < 1. Thus the radius of convergence is 1.
It remains to check convergence when x = −1, 1.
P
(−1)n
x = −1: ∞
converges by the alternating series test.
n=1
P∞ 1n8n−1
P
1
x = 1: n=1 8n−1 diverges by limit comparison test with ∞
n=1 n .
So the interval of convergence is [−1, 1).
April 15, 2015