Taylor Series - PowerPoint by hI14064W

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```									                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!
x
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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