MATH 101 V2A
March 18th – Practice problems
Hints and Solutions
1. Find a power series representation and the radius of convergence for arctan(x). (Hint: First find a
d
power series representation for dx
arctan(x).)
Note: The problem originally asked you to find the interval of convergence, but it should
have said radius.
Solution: Since,
d
dx
arctan(x) =
1
1+x2 ,
and, for |x| < 1,
∞
∞
X
X
1
1
2 n
=
=
(−x ) =
(−1)n x2n .
1 + x2
1 − (−x2 ) n=0
n=0
Therefore
arctan(x) =
∞
X
(−1)n
n=0
x2n+1
2n + 1
for |x| < 1.
2. Determine the radius and interval of convergence for each of the following series.
(a)
∞
X
(−1)n n
x .
n5 5 n
n=1
Solution: We have that
(−1)n+1 xn+1 (n+1)5 5n+1 xn5 (−1)n xn = (n + 1)5 5 ,
5
n
n 5
so, clearly,
(−1)n+1 xn+1 (n+1)5 5n+1 xn5 |x|
=
lim
lim .
= n→∞
55 (−1)n xn
n→∞ (n
+
1)
5
n5 5n
So, the radius of convergence is 5. When x = −5 the series becomes
∞
∞
X
X
(−1)n
1
n
(−5)
=
,
5 5n
5
n
n
n=1
n=1
which converges by the p -Test. When x = 5, the series becomes
∞
∞
X
(−1)n n X (−1)n
5
=
,
n5 5n
n5
n=1
n=1
which converges absolutely by the p -Test, and therefore converges. So, the interval of convergence
is [−5, 5].
∞
X
(1 + 3n )xn
(b)
.
n!
n=0
Hint: Show that this series converges for all x.
2