11.10 Taylor Series         =                =       + +       +     ···              =Question: How (and whether) can we ...
( )=    +       (       )+       (        ) +        (    ) + ···How to determine            ?                        = ( ...
Theorem: If      ( ) has      a power series representationat :    ( )=             (        ) ,         |     |<         ...
The series             ( )          ( )               ( )( )= ( )+        (   )+       (       ) +       (   ) + ···      ...
Ex: Find the Maclaurin series of   ( )=   andits radius of convergence.
Ex: Find the Maclaurin series of   ( )=   andits radius of convergence. ( )= ( )= ( )=    ···
Ex: Find the Maclaurin series of   ( )=   andits radius of convergence. ( )=                  ( )= ( )=                  (...
Ex: Find the Maclaurin series of           ( )=        andits radius of convergence. ( )=                       ( )= ( )= ...
Ex: Find the Maclaurin series of             ( )=        andits radius of convergence. ( )=                         ( )= (...
Ex: Find the Maclaurin series of             ( )=        andits radius of convergence. ( )=                         ( )= (...
Can we say that   =                 ? Not yet.                        =                            !We need to prove that ...
Can we say that         =                         ? Not yet.                               =                              ...
Taylor’s Inequality:     ( + )If           ( )       for   |           |       , then theremainder of the Taylor series sa...
Why   =           ?          =              !
Why     =                 ?            =                      !        ( + )Since           ( )           for   | |
Why      =                 ?               =                       !         ( + )Since            ( )           for   | |...
Why      =                   ?               =                       !         ( + )Since            ( )             for  ...
Why      =                   ?               =                       !         ( + )Since            ( )             for  ...
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Calculus II - 28

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Stewart Calculus 11.10

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  • Calculus II - 28

    1. 1. 11.10 Taylor Series = = + + + ··· =Question: How (and whether) can we representa general function by power series? ( )= + ( )+ ( ) + ( ) + ···How to find , , , ··· ?
    2. 2. ( )= + ( )+ ( ) + ( ) + ···How to determine ? = ( )How to determine ?( )= + ( )+ ( ) + ··· = ( )How to determine ?( )= + · ( )+ · ( ) + ··· = ( )/How to determine ? ( )= · + · · ( ) + ··· = ( )/ !
    3. 3. Theorem: If ( ) has a power series representationat : ( )= ( ) , | |< =then its coefficients are given by the formula ( ) ( ) = !The series ( ) ( ) ( )( )= ( )+ ( )+ ( ) + ( ) + ··· ! ! !is called the Taylor series of the function at .
    4. 4. The series ( ) ( ) ( )( )= ( )+ ( )+ ( ) + ( ) + ··· ! ! !is called the Taylor series of the function at .The special case when = : ( ) ( ) ( ) ( )= ( )+ + + + ··· ! ! !is called the Maclaurin series of the function.
    5. 5. Ex: Find the Maclaurin series of ( )= andits radius of convergence.
    6. 6. Ex: Find the Maclaurin series of ( )= andits radius of convergence. ( )= ( )= ( )= ···
    7. 7. Ex: Find the Maclaurin series of ( )= andits radius of convergence. ( )= ( )= ( )= ( )= ( )= ( )= ··· ···
    8. 8. Ex: Find the Maclaurin series of ( )= andits radius of convergence. ( )= ( )= ( )= ( )= ( )= ( )= ··· ··· ( ) ( ) = = + + + + ···= ! = ! ! ! !
    9. 9. Ex: Find the Maclaurin series of ( )= andits radius of convergence. ( )= ( )= ( )= ( )= ( )= ( )= ··· ··· ( ) ( ) = = + + + + ···= ! = ! ! ! ! + | | = +
    10. 10. Ex: Find the Maclaurin series of ( )= andits radius of convergence. ( )= ( )= ( )= ( )= ( )= ( )= ··· ··· ( ) ( ) = = + + + + ···= ! = ! ! ! ! + | | = + =
    11. 11. Can we say that = ? Not yet. = !We need to prove that . = !
    12. 12. Can we say that = ? Not yet. = !We need to prove that . = ! ( ) ( )Let = ( ) be the n-th degree = !Taylor polynomial of ( ) at . We need toprove ( )= ( ) ( ) .
    13. 13. Taylor’s Inequality: ( + )If ( ) for | | , then theremainder of the Taylor series satisfies theinequality +| ( )| | | , | | ( + )!
    14. 14. Why = ? = !
    15. 15. Why = ? = ! ( + )Since ( ) for | |
    16. 16. Why = ? = ! ( + )Since ( ) for | |by Taylor’s inequality, + | ( )| | | , | | ( + )!
    17. 17. Why = ? = ! ( + )Since ( ) for | |by Taylor’s inequality, + | ( )| | | , | | ( + )! + | | = ( + )!
    18. 18. Why = ? = ! ( + )Since ( ) for | |by Taylor’s inequality, + | ( )| | | , | | ( + )! + | | = ( + )!So ( )= , that is to say, = = !

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