Taylor Series - PowerPoint by hI14064W


									                Brook Taylor
Taylor Series    1685 - 1731

                Ch 12.10
General form…
   Taylor formula provides us with a general
    method for developing power series
    expansions of functions

                         ( n)
                      f     (a)
    f ( x)               n!
                                ( x  a) n

               n 0
Maclaurin Expansion
   This is just a special case of the Taylor
    expansion when a = 0.
                                ( n)
                             f     (0) n
           f ( x)               n!
                      n 0
Where does it come from?
   In Math 205 we showed that the Taylor
    formula can be developed by doing
    integration by parts “the wrong way!” This
    also leads to the general expression for
    the remainder term which can be used to
    estimate the precision of a Taylor
    expansion representation of a function.
Remainder Term…
   If the Taylor series converges and is truncated
    after “n” terms, the error is
                          M          n 1
              Rn ( x)           xa
                        (n  1)!
    where M is the max value of f(n+1)(x)

                                       Examples 12:10: 8,9,46

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