Math 1592 Solutions of Quiz 3
Problem 1. (10 points, 5 each) Determine the convergence of the following sequences.
If convergent, find the limits:
1. an=5n2
n2+2 .
Method 1: Use the L’Hˆopital’s Rule.
lim
n→∞
5n2
n2+ 2 = lim
x→∞
5x2
x2+ 2 = lim
x→∞
10x
2x= 5.
Method 2: Use limit properties.
lim
n→∞
5n2
n2+ 2 = lim
n→∞
5
1 + 2
n2
=5
1 + lim
n→∞
2
n2
= 5.
2. an= (−1)n.
Diverges because the sequence alternates between -1 and 1.
Problem 2. (10 points, 5 each) Determine the convergence of the following series.
If convergent, find their sums:
1.
∞
X
n=0
2µ3
4¶n
.
This is a geometric series. Because 3/4<1, it is convergent and its sum is
∞
X
n=0
2µ3
4¶n
=2
1−3
4
= 8
2.
∞
X
n=1
n2
n2+ 1
Since
lim
n→∞
n2
n2+ 1 = lim
n→∞
1
1 + 1
n2
=1
1 + lim
n→∞
1
n2
= 1 6= 0,
the series diverges.
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