Math 166
Quiz 7
Name:
Directions: This quiz is worth a total of 10 points. To receive full credit, all work must be shown.
1. Use either the direct comparison test or limit comparison test to determine if the series
∞
X
n=1
or diverges where
r
an =
n+1
.
n2 + 5
Your answer should contain these four parts:
1. A statement giving your choice of bn .
2. A statement addressing the convergence or divergence of
P∞
n=1 bn .
3. Either an inequality or a limit computation involving an and bn .
4. A conclusion.
Let
Note that
1
bn = √ .
n
∞
X
bn diverges because it is a p-series with p = 1/2 ≤ 1. Next, note
n=1
r
√
an
n+1
lim
= lim n
n→∞ bn
n→∞
n2 + 5
r
n2 + n
= lim
n→∞
n2 + 5
r
n2 + n
=
lim 2
n→∞ n + 5
= 1.
Therefore,
∞
X
n=1
an converges by the limit comparison test.
an converges
2. Use either the ratio test or root test to determine if the following series converges or diverges
∞
X
(n − 1)!
.
(n
+ 1)2
n=1
Note that
(n + 1)2
an+1
n!
= lim
2
n→∞ an
n→∞ (n + 2) (n − 1)!
n(n + 1)2
= lim
n→∞ (n + 2)2
= ∞.
lim
Therefore, by the ratio test, the series diverges.