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Series Slides

Series Slides. A review of convergence tests. Roxanne M. Byrne University of Colorado at Denver. N th Term Test. This test can be applied to any series. N th Term Test. Lim a n. You must evaluate:. n  . Where { a n } is the sequence of terms of the series. N th Term Test.

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Series Slides

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  1. Series Slides A review of convergence tests Roxanne M. Byrne University of Colorado at Denver

  2. Nth Term Test This test can be applied to any series

  3. Nth Term Test Lim an You must evaluate: n   Where { a n } is the sequence of terms of the series

  4. Nth Term Test If lim a n 0, the series diverges n   Conclusion: If lim a n = 0, the test fails n   Where { a n } is the sequence of terms of the series

  5. Nth Term Test • 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. Remarks: • Conversely, some of the other tests need this • limit evaluated also. Remember this test if the limit is not zero.

  6. Integral Test This test can be applied only to positive term series

  7. 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

  8. INTEGRAL TEST Conclusion: If the integral converges then the series converges If the integral diverges then the series diverges

  9. 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.

  10. Comparison Test This test can be applied only to positive term series

  11. 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

  12. 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.

  13. COMPARISON TEST Remarks: • As with the Nth Term Test, when the test fails, • itmeans 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

  14. Limit Comparison Test This test can be applied only to positive term series

  15. 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 that • has the same end behavior as yours. • If you think it diverges, you must find a • positive termed divergent series that • has the same end behavior as yours. • Evaluate where an and bn are • the terms of your two series

  16. LIMIT COMPARISON TEST Conclusion: • If 0 < < , then both series converge or both series diverge. • If equals zero or increases • without bound or does not exist, then • test fails.

  17. LIMIT COMPARISON TEST Remarks: • When the test fails, you must either find another • comparison series or you must try another test. • The test works well with series that look almost • like 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

  18. Ratio Test This test can be applied only to positive term series

  19. RATIO TEST You must: • Evaluate un + 1 • Evaluate the ratio • Evaluate lim n   Where { u n } is the sequence of terms

  20. 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

  21. RATIO 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 factorials in them.

  22. The N th Root Test This test can be applied only to positive term series

  23. THE NTH ROOT TEST You must: • Find • Evaluate Where { a n } is the sequence of terms of the series

  24. 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

  25. 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 whose • terms have factorials in them.

  26. Absolute Convergence Test This test is used on series with varying signed terms

  27. 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.

  28. ABSOLUTE CONVERGENCE TEST 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.

  29. ABSOLUTE CONVERGENCE TEST 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.

  30. Alternating Series Test This test can be applied only to series that have alternating terms

  31. ALTERNATING SERIES TEST You must: • Make sure the terms are alternating • Define a new sequence, un, as the absolute • value of the terms of your sequence • of terms. • Prove that un is a decreasing sequence. • Evaluate lim un n  

  32. ALTERNATING SERIES TEST Conclusion: If the limit is zero, then alternatingseries converges.

  33. ALTERNATING SERIES TEST Remarks: • 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.

  34. The End PLEASE CLOSE WINDOW

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