9.6
The Ratio and Root Tests
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The Ratio Test
This section begins with a test for absolute
convergence—the Ratio Test.
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Interpreting the Ratio Test
 You know that a series of positive terms converges if and only
if the sequence decreases rapidly towards zero. One way to
measure the rate at which the sequence is decreasing is to
examine the ratio as n grows large.
 Recall that if this ratio in a geometric sequence is less than
one, the sequence converges. Where as if the ratio is greater
than one, the sequence diverges.
t n 1
L

lim
 For the series  t n , if
, then:
n  t
n 1
n

– The series converges if L<1.
– The series diverges if L>1.
– The test is inconclusive when L=1 , so the series may
converge or diverge. Choose another test to confirm.
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Example 1 – Using the Ratio Test
Determine the convergence or divergence of
Solution:
Because an = 2n/n!, you can write the following.
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More Ratio Test Examples
 Using the Ratio Test to determine the convergence or
divergence of each series.

 Example 2 -
 (1)n
n 1
n
n 1
( 1)n 24 n
 Example 3 - 
n 0 (2n  1)!

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The Root Test
Like the ratio test, if L=1, then the series could converge or diverge.
Seek another test to confirm.
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Example 4 – Using the Root Test
Determine the convergence or divergence of
Solution:
You can apply the Root Test as follows.
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More Root Test Examples
 Using the Root Test to determine the convergence or
divergence of each series.

 Example 5 -
 Example 6 -
1

n
(ln
n
)
n 2

1


1  n 

n 0 
n2
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Strategies for Testing Series
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Example 7 – Applying the Strategies for Testing Series
Determine the convergence or divergence of each series.
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Example 7 continued…
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Strategies for Testing Series
12
Strategies for Testing Series
cont’d
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Strategies for Testing Series
cont’d
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