Download Quiz 13 Solutions

Math 2315 - Calculus II
Quiz #13 - 2007.11.28
Solutions
1. Find a power series representation for the function f (x) =
x
4x+1
and determine the interval of convergence.
x
4x + 1
1
=x·
1 + 4x
1
=x·
1 − (−4x)
∞
X
=x·
(−4x)n−1
f (x) =
=x·
n=1
∞
X
(−4)n−1 xn−1
n=1
=
∞
X
1
− (−4x)n−1
4
n=0
This series converges if |−4x| < 1 or − 14 < x < 14 . So the interval of convergence is − 41 , 14 .
2
2. Find a power series representation for the function f (x) = ln(x + 3) and determine the interval of convergence.
So first, we notice that
d
1
ln(x + 3) =
dx
x+3
where
1
1 1
=
x+3
3 1 + x3
1
1
=
3 1 − (− x3 )
∞
1 X x n
.
−
=
3 n=0
3
So we have
∞
1 X x n
−
dx
3 n=0
3
n
∞ 1X
1
1
=
xn+1 + D
−
3 n=0
3
n+1
Z
ln(x + 3) =
=
∞
X
(−1)n x n+1
+D
n+1 3
n=0
Plugging in x = 0 gives that D = ln(3). Therefore
∞
X
(−1)n x n+1
+ ln(3).
ln(x + 3) =
n+1 3
n=0
The interval of convergence can be found from the sum
∞
1 X x n
−
,
3 n=0
3
which requires that − x3 < 1 or −3 < x < 3. So the interval of convergence is (−3, 3).