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Series |
Tests
for convergence |
Ratio
test |
Root
test or Cauchy's root test |
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The
ratio test |
The test
for whether a series is absolutely convergent by testing the limit of the absolute
value of
the ratio of
successive
terms of the series is called the ratio test. |
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(1)
If |
![](SerRT.gif) |
then
the series |
![](SerCompTB.gif) |
is
absolutely
convergent. |
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(2)
If |
![](SerRTA.gif) |
then
the series |
![](SerCompTB.gif) |
is
divergent.
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(3)
If |
![](SerRTB.gif) |
then
the test is inconclusive,
i.e.,
gives
no information. |
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Example: Examine
whether the series |
![](SerRTEx2.gif) |
converges
or diverges. |
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Solution: Since, |
![](SerRTEx2A.gif) |
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the
series converges if 0
< x < 1. |
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Example: Determine
whether the series |
![](SerRTEx3.gif) |
converges
or diverges. |
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Solution: Since, |
![](SerRTEx3A.gif) |
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therefore L
< 1 and the series converges. |
If
the given series has a factorial in the nth
term expression, the ratio test
should be used. |
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Example: Let
show that the logarithmic series |
![](SerRTEx4.gif) |
converges. |
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Solution: |
![](SerRTEx4A.gif) |
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thus, the series
converges
for x < 1.
The sum of the series |
![](SerRTEx4B.gif) |
|
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Root
test or Cauchy's root test |
The test
for whether a series |
![](SerCompTB.gif) |
of
positive terms is absolutely
convergent by considering; |
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(1)
If |
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then
the series |
![](SerCompTB.gif) |
is
absolutely
convergent. |
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(2)
If |
![](SerRtTA.gif) |
then
the series |
![](SerCompTB.gif) |
is
divergent.
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(3)
If |
![](SerRtTB.gif) |
then
the test is inconclusive,
i.e.,
gives
no information. |
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Example: Determine
whether the series |
![](SerRootEx1.gif) |
converges
or diverges. |
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Solution: The
given geometric series converges since |
![](SerRootEx1A.gif) |
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Example: Let
show that the series |
![](SerRootEx2.gif) |
converges. |
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Solution: The
series converges since |
![](SerRootEx2A.gif) |
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Example: Test
the series |
![](SerRootEx3.gif) |
for
convergence. |
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Solution: The
series converges since |
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We
use
the root test when the entire nth
term expression is raised to some power of n. |
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