11.3 The Integral TestThere are many series that cannot be easilyevaluated. We need a method to  determine if it is conver...
The integral test:Suppose ( ) is a continuous positivedecreasing function and let   = ( ) , thenthe series                ...
Improper integral of Type I: If       ( )   exists for every         , then                ( )   =            ( )provided ...
Ex: Determine if series           is convergent.                          =                              +
Ex: Determine if series            is convergent.                          =                              +    Let   ( )= ...
Ex: Determine if series            is convergent.                          =                              +    Let   ( )= ...
Ex: Determine if series            is convergent.                          =                              +    Let   ( )= ...
Ex: Determine if series            is convergent.                          =                              +    Let   ( )= ...
Ex: Determine if series                is convergent.                          =                                +    Let  ...
Ex: Determine if series               is convergent.                           =                                 +    Let ...
Ex: Determine if series       is convergent.                          =
Ex: Determine if series       is convergent.                          =    Let   ( )=
Ex: Determine if series           is convergent.                          =    Let   ( )=    it is continuous, positive bu...
Ex: Determine if series           is convergent.                          =    Let   ( )=    it is continuous, positive bu...
Ex: Determine if series           is convergent.                          =    Let   ( )=    it is continuous, positive bu...
Ex: Determine if series                   is convergent.                              =    Let   ( )=    it is continuous,...
Ex: Determine if series                   is convergent.                              =    Let   ( )=    it is continuous,...
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 c...
Ex: Estimate the sum of the series                                     =    using   =   .
Ex: Estimate the sum of the series                                     =    using   =     .     ( )=       is continuous p...
Ex: Estimate the sum of the series                                     =    using   =     .     ( )=       is continuous p...
Ex: Estimate the sum of the series                                           =    using       =       .     ( )=          ...
Ex: Estimate the sum of the series                                              =    using       =       .     ( )=       ...
Ex: Estimate the sum of the series                                                  =    using       =       .     ( )=   ...
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Calculus II - 23

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Stewart Calculus Section 11.3

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  • Calculus II - 23

    1. 1. 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: = +
    2. 2. The integral test:Suppose ( ) is a continuous positivedecreasing function and let = ( ) , thenthe series =is convergent if and only if the improperintegral ( )is convergent.
    3. 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. 4. Ex: Determine if series is convergent. = +
    5. 5. Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing.
    6. 6. Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing. = + +
    7. 7. Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing. = + + =
    8. 8. Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing. = + + = =
    9. 9. Ex: Determine if series is convergent. = + Let ( )= + it is continuous, positive and decreasing. = + + = = So is convergent. = +
    10. 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. 11. Ex: Determine if series is convergent. =
    12. 12. Ex: Determine if series is convergent. = Let ( )=
    13. 13. Ex: Determine if series is convergent. = Let ( )= it is continuous, positive but not decreasing.
    14. 14. Ex: Determine if series is convergent. = Let ( )= it is continuous, positive but not decreasing. however, it is decreasing when > , fine.
    15. 15. Ex: Determine if series is convergent. = Let ( )= it is continuous, positive but not decreasing. however, it is decreasing when > , fine. =
    16. 16. Ex: Determine if series is convergent. = Let ( )= it is continuous, positive but not decreasing. however, it is decreasing when > , fine. = ( ) = =
    17. 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. 18. Ex: p-test is convergent if > , otherwise divergent.
    19. 19. Ex: p-test is convergent if > , otherwise divergent. is convergent if > , otherwise divergent.=
    20. 20. Reminder estimate:Suppose ( ) is a continuous positivedecreasing function and let = ( ). ∞If = = is convergent, then + ( ) + ( ) +or ( ) ( ) +where = is called the remainder.
    21. 21. Ex: Estimate the sum of the series = using = .
    22. 22. Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing
    23. 23. Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing + +
    24. 24. Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing + + + + · ·
    25. 25. Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing + + + + · · = + + ··· + .
    26. 26. Ex: Estimate the sum of the series = using = . ( )= is continuous positive decreasing + + + + · · = + + ··· + . . .

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