Worksheet 10/31/08
Find the radius of convergence and the interval of convergence
Problem 1. โ
X
n=2
(โ1)nxn
4 ln(n)
Solution.
lim
nโโ ๎๎๎๎
(โ1)n+1xn+14 ln n
(โ1)nxn4 ln(n+ 1)๎๎๎๎
= lim
nโโ
|x|ln n
ln(n+ 1) =|x|
We know it to be absolutely convergent if
โ |x|<1
Checking the endpoints we see that it converges at x= 1 (alternating series test), but not
at x=โ1 (comparison test).
Thus, the interval of convergence is (โ1,1].
Problem 2. โ
X
n=1
nnxn
Solution. If x6= 0, then
lim
nโโ
n
p|nnxn|= lim
nโโ
nx =โ
Thus, this series converges only when x= 0.
