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Ratio and Root Tests for Convergence in Series

Learn how to apply Ratio Test & Root Test to determine if series converge/diverge, along with guidelines on when to use each test. Understand the conditions and application of both tests to ensure accurate results. Discover the secrets and steps to apply different tests like Direct Comparison, Limit Comparison, Integral Test, and more. Master the techniques for convergence analysis in series.

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Ratio and Root Tests for Convergence in Series

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  1. Chapter 9.6 THE RATIO AND ROOT TESTS

  2. 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!

  3. THEOREM 9.17RATIO TEST Condition: Let be a series with nonzero terms. • converges absolutely if • diverges if or • The Ratio Test is inconclusive if

  4. USING THE RATIO TEST Determine the convergence or divergence of Does this series have nonzero terms?

  5. You got it! All terms are nonzero…that’s a pretty easy condition to fulfill, don’t you think? Now we need to find and

  6. So let’s make a ratio!

  7. Conclusion: By the ratio test, this is an absolutely convergent series, and therefore convergent. Now for the test…

  8. THEOREM 9.18ROOT TEST Condition: Let be a series. • converges absolutely if • diverges if or • The Root Test is inconclusive if

  9. USING THE ROOT TEST Determine the convergence or divergence of Is this a series?

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

  11. Conclusion: By the root test, this is an absolutely convergent series, and therefore convergent. Now for the test…

  12. WHICH TEST TO APPLY WHEN?! Shhh…Let’s talk about the series secrets

  13. STEP 1 Apply the nth Term Test. If , the infinite series diverges and you’re all done!

  14. STEP 2 Check if the series is a geometric series or a p-series. • If it’s a geometric series with then it converges. • If it is a p-series with then the series converges.

  15. 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.

  16. 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.

  17. 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 .

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