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Calculus II - 25

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Stewart Calculus Section 11.5&6

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Calculus II - 25

  1. 1. 11.5 Alternating Series11.6 The Ratio and Root Tests An alternating series is a series whose terms are alternately positive and negative. For example: ( ) = + + ··· =
  2. 2. The Alternating Series Test:If the alternating series ( ) = + ··· > =satisfies + and =then the series is convergent.
  3. 3. ( )Ex: Test series for convergence. = +
  4. 4. ( )Ex: Test series for convergence. = + = > +
  5. 5. ( )Ex: Test series for convergence. = + = > + =
  6. 6. ( )Ex: Test series for convergence. = + = > + = decreasing?
  7. 7. ( )Ex: Test series for convergence. = + = > + = decreasing?When , it is decreasing (check by derivative).
  8. 8. ( )Ex: Test series for convergence. = + = > + = decreasing?When , it is decreasing (check by derivative). ( ) So is convergent. = +
  9. 9. Notice that = + + + + ··· =is divergent but ( ) = + + ··· =is convergent. ∞A series = is called absolutely ∞convergent if = | | is convergent.It is called conditionally convergent if it is ∞convergent but = | | is divergent.
  10. 10. Theorem: Absolutely convergent series is convergent. | | | | < is convergent because <= | | | |
  11. 11. Theorem: Absolutely convergent series is convergent. Ex: Determine if is convergent. = | | | | < is convergent because <= | | | |
  12. 12. Theorem: Absolutely convergent series is convergent. Ex: Determine if is convergent. = It has both positive and negative terms, but not alternating. | | | | < is convergent because <= | | | |
  13. 13. Theorem: Absolutely convergent series is convergent. Ex: Determine if is convergent. = It has both positive and negative terms, but not alternating. | | | | < is convergent because <= | | | | so is absolutely convergent, =
  14. 14. Theorem: Absolutely convergent series is convergent. Ex: Determine if is convergent. = It has both positive and negative terms, but not alternating. | | | | < is convergent because <= | | | | so is absolutely convergent, = so is convergent. =
  15. 15. The Ratio Test: +If = < , then the series =is absolutely convergent (and thus convergent). +If = > or = , then the series is divergent. = +If = , no conclusion can be drawn.
  16. 16. The Root Test:If | |= < , then the series =is absolutely convergent (and thus convergent).If | |= > or = , then the series is divergent. =If | |= , no conclusion can be drawn.
  17. 17. Ex: Test the series for convergence. =
  18. 18. Ex: Test the series for convergence. = + ( + ) + = + = So is (absolutely) convergent. =
  19. 19. +Ex: Test the series for convergence. = +
  20. 20. +Ex: Test the series for convergence. = + + | |= + + So is (absolutely) convergent. = +

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