series slides a review of convergence tests roxanne m. byrne university of colorado at denver
TRANSCRIPT
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Series Slides
A review of convergence tests
Roxanne M. ByrneUniversity of Colorado at Denver
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Nth Term Test
This test can be applied to any series
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Nth Term Test
You must evaluate:
Lim an n
Where { a n } is the sequence of terms of the series
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Nth Term Test
Conclusion:
n If lim a n 0, the series diverges
If lim a n = 0, the test failsn
Where { a n } is the sequence of terms of the series
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Nth Term Test
Remarks:• Remember, if the limit is zero,
THE TEST FAILS. This means you must try a different test.
• Sometimes the limit is not easy to evaluate. In this case, try other test that you think might be more productive first.
• Conversely, some of the other tests need thislimit evaluated also. Remember this test if the limit is not zero.
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Integral Test
This test can be applied only to positive term series
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INTEGRAL TEST
You must:
• Find a continuous function, f(x), such that f(n) = an
• Verify that f(x) is a decreasing function
• Determine if f(x) dx converges
Where { a n } is the sequence of terms of the series
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INTEGRAL TEST
Conclusion:
If the integral converges
then the series converges
If the integral diverges
then the series diverges
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INTEGRAL TEST
Remarks:
• This is both a convergence and divergence test
• If f(x) is an increasing function, go to the
Nth Term Test.
• This test requires that the function can be
integrated. It will not work for series whose
terms have factorials in them.
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Comparison Test
This test can be applied only to positive term series
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COMPARISON TEST
You must:
• Decide if you think the series converges or
diverges
• If you think it converges, you must find a
larger termed series that you know
converges.
• If you think it diverges, you must find a
smaller positive termed series that you
know diverges
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COMPARISON TEST
Conclusion:
• If you find a larger termed convergent series,
then your series converges.
• If you find a smaller positive termed divergent
series, then your series diverges.
• If you cannot find an appropriate comparison
series, the test fails.
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COMPARISON TEST
Remarks:• As with the Nth Term Test, when the test fails,
it means you must try another test.
• The test works well with series that look almost like a geometric series or a p-series.
• The major disadvantages of this test: You must decide beforehand if the series converges or diverges. You must find a corresponding
comparison series
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Limit Comparison Test
This test can be applied only to positive term series
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LIMIT COMPARISON TEST
You must:• Decide if you think the series converges or
diverges
• If you think it converges, you must find a positive termed convergent series thathas the same end behavior as yours.
• If you think it diverges, you must find a positive termed divergent series thathas the same end behavior as yours.
• Evaluate where an and bn are
the terms of your two series
n
n
n a
blim
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• If 0 < < , then both series
converge or both series
diverge.
• If equals zero or increases
without bound or does not exist, then
test fails.
LIMIT COMPARISON TEST
n
n
n a
blim
Conclusion:
n
n
n a
blim
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LIMIT COMPARISON TEST
• When the test fails, you must either find anothercomparison series or you must try another test.
• The test works well with series that look almostlike a geometric series or p-series.
• The major disadvantages of this test: You must decide beforehand if the series
converges or diverges. You must find a corresponding
comparison series
Remarks:
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Ratio Test
This test can be applied only to positive term series
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You must:
• Evaluate u n + 1
• Evaluate the ratio
• Evaluate lim
RATIO TEST
n
n
u
u 1
n
n
u
u 1
n
Where { u n } is the sequence of terms
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RATIO TEST
Conclusion:
If the limit < 1 then
the series converges
If the limit > 1 then
the series diverges
If the limit = 1 then
the test fails
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Remarks:
• This is both a convergence and divergence test
• This test can be used to prove absolute
convergence
• This test will not work on series whose terms
are rational functions of n. For these
series, use the Limit Comparison Test
and the end behavior of the terms.
RATIO TEST
• This test works well with series whose terms have factorials in them.
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The N th Root Test
This test can be applied only to positive term series
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THE NTH ROOT TEST
You must:
• Find
• Evaluate
nna
nn
nalim
Where { a n } is the sequence of terms of the series
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THE NTH ROOT TEST
Conclusion:
If the limit < 1 then
the series converges
If the limit > 1 then
the series diverges
If the limit = 1 then
the test fails
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THE NTH ROOT TEST
Remarks:• This is both a convergence and divergence test
• This test can be used to prove absolute
convergence • This test will not work on series whose terms are
rational functions of n. For these series,
use the Limit Comparison Test and the end
behavior of the terms.
• This test works well with series whose terms have powers of n in them.
• This test does not work well with series whoseterms have factorials in them.
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Absolute Convergence Test
This test is used on series with varying signed terms
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ABSOLUTE CONVERGENCE TEST
You must:
• Let bn be the absolute value of the
sequence of terms of your series
• Determine if the sum of bn is a convergent
series by one of the positive term
convergence tests.
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Conclusion:
• If the sum of bn converges then the
original series converges absolutely
• If the sum of bn converges then the
original series converges conditionally
or it diverges.
ABSOLUTE CONVERGENCE TEST
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Remarks:
• If the sum of bn diverges then you usually
use the alternating series test to
determine if the original series converges.
• If you want to determine the type of
convergence of an alternating series,
you would use this test first.
ABSOLUTE CONVERGENCE TEST
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Alternating Series Test
This test can be applied only to series that have
alternating terms
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You must:
• Make sure the terms are alternating
• Define a new sequence, un, as the absolute value of the terms of your sequenceof terms.
• Prove that un is a decreasing sequence.
• Evaluate lim un
ALTERNATING SERIES TEST
n
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ALTERNATING SERIES TEST
Conclusion:
If the limit is zero,
then alternating series
converges.
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ALTERNATING SERIES TEST
• If you need to determine if the series is absolutely
or conditionally convergent, you must test to
see if un converges using a positive term
series test.
• If the lim un 0 or if un is an increasing
sequence, use the N th Term Test.
Remarks:
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The EndThe End
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