11.8 Power SeriesA power series is a series of the form           =    +     +       +      + ···   ={ }    are called the...
Ex: For what values of   is the series    convergent?                          =
Ex: For what values of   is the series    convergent?                          =It is a geometric series. When   <   <   ,...
Ex: For what values of   is the series       !    convergent?                          =
Ex: For what values of    is the series       !    convergent?                           =                    +   +     ( ...
Ex: For what values of    is the series       !    convergent?                           =                    +   +     ( ...
Ex: For what values of    is the series        !    convergent?                           =                    +   +     (...
Ex: For what values of   is the series    convergent?                          =
Ex: For what values of   is the series    convergent?                          =          | | |   |=
Ex: For what values of     is the series    convergent?                            =           | | |   |=       |    |=   ...
Ex: For what values of     is the series    convergent?                            =           | |  |   |=       |    |=  ...
(   )Ex: For what values of   is the series    convergent?                          =
(   )Ex: For what values of       is the series    convergent?                              =                    +   +    ...
(   )Ex: For what values of            is the series    convergent?                                   =                   ...
(       )Ex: For what values of            is the series    convergent?                                   =               ...
(       )Ex: For what values of             is the series    convergent?                                       =          ...
∞A series of the form   =     ( − )   is called apower series centered at   or a power series about .
∞A series of the form    =     ( − )   is called apower series centered at    or a power series about .                   ...
Ex: Find the radius of convergence of the series                         ( + )                            +               ...
Ex: Find the radius of convergence of the series                          ( + )                              +            ...
Ex: Find the radius of convergence of the series                             ( + )                                 +      ...
Ex: Find the radius of convergence of the series                             ( + )                                 +      ...
Ex: Find the radius of convergence of the series                             ( + )                                 +      ...
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Calculus II - 26

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Stewart Calculus 11.8

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  • Calculus II - 26

    1. 1. 11.8 Power SeriesA power series is a series of the form = + + + + ··· ={ } are called the coefficient of the series.For each fixed , the series can beconvergent or not.The sum of the series is a function of .
    2. 2. Ex: For what values of is the series convergent? =
    3. 3. Ex: For what values of is the series convergent? =It is a geometric series. When < < , the series converges.
    4. 4. Ex: For what values of is the series ! convergent? =
    5. 5. Ex: For what values of is the series ! convergent? = + + ( + )! = = ( + )| | !
    6. 6. Ex: For what values of is the series ! convergent? = + + ( + )! = = ( + )| | ! + = ( + )| | = except =
    7. 7. Ex: For what values of is the series ! convergent? = + + ( + )! = = ( + )| | ! + = ( + )| | = except =The series converges when = .
    8. 8. Ex: For what values of is the series convergent? =
    9. 9. Ex: For what values of is the series convergent? = | | | |=
    10. 10. Ex: For what values of is the series convergent? = | | | |= | |= for any
    11. 11. Ex: For what values of is the series convergent? = | | | |= | |= for anyThe series converges for any .
    12. 12. ( )Ex: For what values of is the series convergent? =
    13. 13. ( )Ex: For what values of is the series convergent? = + + ( ) ( ) = = | | + +
    14. 14. ( )Ex: For what values of is the series convergent? = + + ( ) ( ) = = | | + + + =| |
    15. 15. ( )Ex: For what values of is the series convergent? = + + ( ) ( ) = = | | + + + =| |The series converges when | |< i.e. < <
    16. 16. ( )Ex: For what values of is the series convergent? = + + ( ) ( ) = = | | + + + =| |The series converges when | |< i.e. < <In fact, it is convergent when < .
    17. 17. ∞A series of the form = ( − ) is called apower series centered at or a power series about .
    18. 18. ∞A series of the form = ( − ) is called apower series centered at or a power series about . ∞Theorem: For a given power series = ( − )there are only three possibilities: The series converges only when = . The series converges for all . The series converges when | |< and diverges when | |> is called the radius of convergence. Anything can happen = ± .
    19. 19. Ex: Find the radius of convergence of the series ( + ) + =
    20. 20. Ex: Find the radius of convergence of the series ( + ) + = + + ( + )( + ) ( + ) + | + | = + + =
    21. 21. Ex: Find the radius of convergence of the series ( + ) + = + + ( + )( + ) ( + ) + | + | = + + = + | + | =
    22. 22. Ex: Find the radius of convergence of the series ( + ) + = + + ( + )( + ) ( + ) + | + | = + + = + | + | = The series converges when | + |< i.e. < <
    23. 23. Ex: Find the radius of convergence of the series ( + ) + = + + ( + )( + ) ( + ) + | + | = + + = + | + | = The series converges when | + |< i.e. < < The radius of convergence is .

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