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11.3 The Integral Test
There are many series that cannot be easily
evaluated. 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 positive
decreasing function and let   = ( ) , then
the series


                  =

is convergent if and only if the improper
integral

                      ( )

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 positive
decreasing 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


       +                                  +

            +                             +
                    ·                         ·
       =        +           + ··· +           .

            .                         .

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Calculus II - 23

  • 1. 11.3 The Integral Test There are many series that cannot be easily evaluated. We need a method to determine if it is convergent without knowing the precise quantity. estimate the sum approximately. Ex: = +
  • 2. The integral test: Suppose ( ) is a continuous positive decreasing function and let = ( ) , then the series = is convergent if and only if the improper integral ( ) is convergent.
  • 3. 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 ∞
  • 4. Ex: Determine if series is convergent. = +
  • 5. Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing.
  • 6. Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing. = + +
  • 7. Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing. = + + =
  • 8. Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing. = + + = =
  • 9. Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing. = + + = = So is convergent. = +
  • 10. 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
  • 11. Ex: Determine if series is convergent. =
  • 12. Ex: Determine if series is convergent. = Let ( )=
  • 13. Ex: Determine if series is convergent. = Let ( )= it is continuous, positive but not decreasing.
  • 14. Ex: Determine if series is convergent. = Let ( )= it is continuous, positive but not decreasing. however, it is decreasing when > , fine.
  • 15. Ex: Determine if series is convergent. = Let ( )= it is continuous, positive but not decreasing. however, it is decreasing when > , fine. =
  • 16. Ex: Determine if series is convergent. = Let ( )= it is continuous, positive but not decreasing. however, it is decreasing when > , fine. = ( ) = =
  • 17. Ex: Determine if series is convergent. = Let ( )= it is continuous, positive but not decreasing. however, it is decreasing when > , fine. = ( ) = = So is divergent. =
  • 18. Ex: p-test is convergent if > , otherwise divergent.
  • 19. Ex: p-test is convergent if > , otherwise divergent. is convergent if > , otherwise divergent. =
  • 20. Reminder estimate: Suppose ( ) is a continuous positive decreasing function and let = ( ). ∞ If = = is convergent, then + ( ) + ( ) + or ( ) ( ) + where = is called the remainder.
  • 21. Ex: Estimate the sum of the series = using = .
  • 22. Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing
  • 23. Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing + +
  • 24. Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing + + + + · ·
  • 25. Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing + + + + · · = + + ··· + .
  • 26. Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing + + + + · · = + + ··· + . . .

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