THE RATIO AND ROOT TESTS

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Chapter 9.6

• THE RATIO AND ROOT TESTS

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After you finish your HOMEWORK you will be able
to

• Use the Ratio Test to determine whether a series
  converges or diverges
• Use the Root Test to determine whether a series
  converges or diverges.
• Know when to use what test!

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THEOREM 9.17RATIO TEST

• Condition Let be a series with nonzero
  terms.
• converges absolutely if
• diverges if or
• The Ratio Test is inconclusive if

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USING THE RATIO TEST

• Determine the convergence or divergence of
• Does this series have nonzero terms?

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• You got it! All terms are nonzerothats a pretty
  easy condition to fulfill, dont you think?
• Now we need to find and

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• So lets make a ratio!

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Now for the test

• Conclusion
• By the ratio test, this is an absolutely
  convergent series, and therefore convergent.

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THEOREM 9.18ROOT TEST

• Condition Let be a series.
  
• converges absolutely if
• diverges if or
• The Root Test is inconclusive if

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USING THE ROOT TEST

• Determine the convergence or divergence of
• Is this a series?

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• Sure is! I like this condition best of all!
• Now we need to find and

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Now for the test

• Conclusion
• By the root test, this is an absolutely
  convergent series, and therefore convergent.

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WHICH TEST TO APPLY WHEN?!

• ShhhLets talk about the series secrets

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STEP 1

• Apply the nth Term Test.
• If , the infinite series
• diverges and youre all done!

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STEP 2

• Check if the series is a geometric series or a
  p-series.
• If its a geometric series with then
  it converges.
• If it is a p-series with then the
  series converges.

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STEP 3

• If is a positive term series, use one of
  the following tests.
• Direct Comparison Test
• Limit Comparison Test
• For these tests we usually compare with a
  geometric or p-series.

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STEP 3 CONT.

• 3. Ratio Test When there is an n!, or a .
• 4. Root Test When there is an or some
  function of n to the nth power.
• 5. Integral Test Must have terms that are
  decreasing toward zero and that can be integrated.

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STEP 4

• If is an alternating series, use one of
  the following tests.
• Alternating Series Test.
• A test from step 3 applied to .
• If so does .