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Lecture 22

MATA37H3 Lecture Notes - Lecture 22: Alternating Series Test, Ratio TestPremium


Department
Mathematics
Course Code
MATA37H3
Professor
Smith( G)
Lecture
22

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MATA37 - Lecture 22 - Alternating Series Test, Ratio Test, and Root Test
Alternating Series Test
• Given
X
n=1
(1)n+1an, an0, if:
1. anan+1nN,and
2. lim
n→∞
an= 0
then
X
n=1
(1)n+1anconverges
Example 1: Does
X
n=2
3(1)n
ln(n)converge or diverge?
an=3
ln(n), an+1 =3
ln(n+ 1)
lim
n→∞
an= lim
n→∞
3
ln(n)= 0,#2 is true
For nN, n 2,ln(n)ln(n+ 1),because ln(x)is increasing on (0,)
=1
ln(n)1
ln(n+ 1) 3
ln(n)3
ln(n+ 1)
Note 3
ln(n)=an,3
ln(n+ 1) =an+1,#1 is true
by alternating series test, our series
X
n=2
3(1)n
ln(n)converges
Ratio Test
See page 633
• Suppose
X
n=1
anis positive termed, i.e. an>0nN. Define lim
n→∞
an+1
an
=L0
1. L < 1 =
X
n=1
anconverges
2. L > 1 =
X
n=1
andiverges
3. L= 1 =test is inconclusive
Case #1: use PMI and CT (to a geometric series)
1
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