18. Taylors Theorem
Taylors Theorem: Su ciently smooth functions
can be approximated locally by polynomials. Sup-
pose ( ) has derivatives on ( ) with ( )
continuous on [ ]. (We put (0( ) = ( );
the assumptions imply existence and continuity of
( ( ) on ( ) for .) Then for [ ]
there is a point between and such that
X ( ) ( ) ( ) ( )
( ) = ( ) ! + ( ) !
What does this say when = 1?
Example: ( ) = ; expand around = 0
( ( ) = so that
(0) = 1 94
Then for some between 0 and (i.e. | | | |)
= ( )(0) + ( )( )
=0 ! !
= !+ !
Write this as
= ( ) + ( )
We can do this since is a, generally quite compli-
cated, function of . If ( ) 0 as we
say that the series lim ( ) = =1 !
represents the function .
It is indeed true in this example tha( )
0 as . We will later show that it does
so in a nice uniform way; for now it is enough
to show that it does so for any