MATH 106A - CALCULUS II
WINTER 2007
QUIZ 9
NAME:
Show ALL your work CAREFULLY.
(a) Find the radius of convergence for the following power series.
โ
X
k=0
x2k
k!
Using the Ratio Test, we have
lim
nโโ
|x|2(n+1)
(n+ 1)! ยทn!
|x|2n= lim
nโโ
|x|2
n+1 =0<1.
It follows that the series converges absolutely for ALL x. This means
that the radius of convergence is infinite.
(b) Determine whether the following alternating series converges absolutely or
conditionally, or diverges. If it converges, find upper and lower bounds for its
limit. โ
X
n=0
(โ1)nn2
5n
Consider
lim
nโโ
(n+1)
2
5n+1 ยท5n
n2= lim
nโโ ๎n+1
n๎2
ยท1
5=1
5<1.
It follows that the series converges absolutely. The series is alternating
so the limit Slies between any two consecutive partial sums Snand
Sn+1. Here, S1=0and S1=โ1
5. Thus, โ0.2<S<0. If we use S2=
โ1
5+4
25 =โ1
25 =โ0.04 and S3=S2โ9
125 =โ0.072, then we can conclude
that โ0.072 <S<โ0.04.
Date: March 30, 2007.
1