Quiz 2 Solutions
Problem 1 (5 points): Determine if the series
1
X
n=0
sin2(n)
4nconverges or di-
verges. Give reasons for your answer.
Since 0sin2(n)
4n1
4n, and
1
X
n=0
1
4nconverges, being a geometric series, it
follows from the direct comparison test that
1
X
n=0
sin2(n)
4nconverges.
Problem 2 (5 points): Determine if the series
1
X
n=1
7(ln n)n
nnconverges or di-
verges. Give reasons for your answer.
By the root test,
1
X
n=1
7(ln n)n
nnconverges if limn!1 7(lnn)n
nn1
n= limn!1 71
nln n
n<
1. But limn!1 71
n= 70= 1 and limn!1 ln n
n= limn!1
1
n
1= 0, so that
limn!1 71
nln n
n= 0 <1. Therefore,
1
X
n=1
7(ln n)n
nnconverges.
1
