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Calculus II - 23
 

Calculus II - 23

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Stewart Calculus Section 11.3

Stewart Calculus Section 11.3

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Calculus II - 23 Calculus II - 23 Presentation Transcript

  • 11.3 The Integral TestThere are many series that cannot be easilyevaluated. We need a method to determine if it is convergent without knowing the precise quantity. estimate the sum approximately.Ex: = +
  • The integral test:Suppose ( ) is a continuous positivedecreasing function and let = ( ) , thenthe series =is convergent if and only if the improperintegral ( )is convergent.
  • Improper integral of Type I: If ( ) exists for every , then ( ) = ( )provided this limit exists as a finite number.We call this improper integral convergent, otherwise divergent. a ∞
  • Ex: Determine if series is convergent. = +
  • Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing.
  • Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing. = + +
  • Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing. = + + =
  • Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing. = + + = =
  • Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing. = + + = = So is convergent. = +
  • Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing. = + + = = it is! So is convergent. what + know don’t ut we = B
  • Ex: Determine if series is convergent. =
  • Ex: Determine if series is convergent. = Let ( )=
  • Ex: Determine if series is convergent. = Let ( )= it is continuous, positive but not decreasing.
  • Ex: Determine if series is convergent. = Let ( )= it is continuous, positive but not decreasing. however, it is decreasing when > , fine.
  • Ex: Determine if series is convergent. = Let ( )= it is continuous, positive but not decreasing. however, it is decreasing when > , fine. =
  • Ex: Determine if series is convergent. = Let ( )= it is continuous, positive but not decreasing. however, it is decreasing when > , fine. = ( ) = =
  • Ex: Determine if series is convergent. = Let ( )= it is continuous, positive but not decreasing. however, it is decreasing when > , fine. = ( ) = = So is divergent. =
  • Ex: p-test is convergent if > , otherwise divergent.
  • Ex: p-test is convergent if > , otherwise divergent. is convergent if > , otherwise divergent.=
  • Reminder estimate:Suppose ( ) is a continuous positivedecreasing function and let = ( ). ∞If = = is convergent, then + ( ) + ( ) +or ( ) ( ) +where = is called the remainder.
  • Ex: Estimate the sum of the series = using = .
  • Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing
  • Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing + +
  • Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing + + + + · ·
  • Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing + + + + · · = + + ··· + .
  • Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing + + + + · · = + + ··· + . . .