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SERIES TESTS

SERIES TESTS. Question in the exam. Is the series convergent or divergent?. Special Series:. Series Tests. Geometric Series Harmonic Series Telescoping Series Alter Harmonic p-series Alternating p-series. Divergence Test Integral Test Comparison Test Limit Compar Test

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SERIES TESTS

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  1. SERIES TESTS Question in the exam Is the series convergent or divergent? Special Series: Series Tests • Geometric Series • Harmonic Series • Telescoping Series • Alter Harmonic • p-series • Alternating p-series • Divergence Test • Integral Test • Comparison Test • Limit Compar Test • Ratio Test • Root Test • Alter Series Test

  2. ALTERNATING SERIES series with positive terms Series Tests • Divergence Test • Integral Test • Comparison Test • Limit Compar Test • Ratio Test • Root Test • Alter Series Test series with some positive and some negative terms All alternating series

  3. ALTERNATING SERIES alternating series n-th term of the series are positive

  4. ALTERNATING SERIES alternating series alternating harmonic series alternating geomtric series alternating p-series

  5. ALTERNATING SERIES THEOREM: (THE ALTERNATING SERIES TEST) alternating decreasing lim = 0 convg Remark: The convergence tests that we have looked at so far apply only to series with positive terms. In this section and the next we learn how to deal with series whose terms are not necessarily positive. Of particular importance are alternating series, whose terms alternate in sign. Example: Determine whether the series converges or diverges.

  6. ALTERNATING SERIES THEOREM: (THE ALTERNATING SERIES TEST) alternating decreasing lim = 0 convg Example: Determine whether the series converges or diverges.

  7. ALTERNATING SERIES THEOREM: (THE ALTERNATING SERIES TEST) alternating decreasing lim = 0 convg Example: Determine whether the series converges or diverges.

  8. Alternating Series, Absolute and Conditional Convergence the series of absolute values DEF: Example: The series Test the series for absolute convergence. is called Absolutely convergent (AC) If the series of absolute values is convergent Also we may say that converges absolutely

  9. Alternating Series, Absolute and Conditional Convergence DEF: IF Is called Absolutely convergent convergent converges absolutely Example: Test the series for absolute convergence.

  10. Alternating Series, Absolute and Conditional Convergence DEF: The series Conditionally convergent (CC) is called If it is convergent but the series of absolute values is divergent DEF: Example: Is called conditionally convergent Test the series for absolute convergence. if it is convergent but not absolutely convergent. REM: convg divg

  11. Alternating Series, Absolute and Conditional Convergence Absolutely convergent convergent THM: convg convg THM: Example: Determine whether the series converges or diverges. The signs change irregularly

  12. Alternating Series, Absolute and Conditional Convergence conditionally convergent Absolutely convergent convergent divergent

  13. SERIES TESTS Questions in the exam Is the series convergent or divergent? Special Series: • Geometric Series • Harmonic Series • Telescoping Series • Alter Harmonic • p-series • Alternating p-series

  14. ALTERNATING SERIES series with positive terms Series Tests • Divergence Test • Integral Test • Comparison Test • Limit Compar Test • Ratio Test • Root Test • Alter Series Test series with some positive and some negative terms All alternating series

  15. THE RATIO AND ROOT TESTS Ratio Test be an infinite series the Ratio Test is inconclusive; that is, no conclusion can be drawn about the convergence or divergence Example: Example: Test the series for convergence. Test the series for convergence.

  16. THE RATIO AND ROOT TESTS TERM-101

  17. THE RATIO AND ROOT TESTS Root Test be an infinite series Example: Example: Test the series for convergence. Test the series for convergence.

  18. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS TERM-082

  19. SERIES TESTS Question in the exam Is the series convergent or divergent? Special Series: Series Tests • Geometric Series • Harmonic Series • Telescoping Series • Alter Harmonic • p-series • Alternating p-series • Divergence Test • Integral Test • Comparison Test • Limit Compar Test • Ratio Test • Root Test • Alter Series Test

  20. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS TERM-091

  21. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS TERM-082

  22. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS TERM-082

  23. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS TERM-091

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