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Stewart Calculus Section 11.3

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- 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. The integral test:Suppose ( ) is a continuous positivedecreasing function and let = ( ) , thenthe series =is convergent if and only if the improperintegral ( )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 positivedecreasing 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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