Class Notes (838,814)
Canada (511,109)
Statistics (248)
STAT312 (22)
Lecture

Taylor’s Theorem.pdf

4 Pages
133 Views
Unlock Document

Department
Statistics
Course
STAT312
Professor
Douglas Wiens
Semester
Fall

Description
93 18. Taylor’s Theorem • Taylor’s Theorem: ‘Su ciently smooth functions can be approximated locally by polynomials.’ Sup- ( 1) 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 1 X ( ) ( ) ( ) ( ) ( ) = ( ) ! + ( ) ! =0 What does this say when = 1? • Example: ( ) = ; expand around = 0 (‘Maclaurin series’): ( ( ) = so that ( ) (0) = 1 94 Then for some between 0 and (i.e. | | | |) X 1 = ( )(0) + ( )( ) =0 ! ! 1 X = !+ ! =0 Write this as = ( ) + ( ) We can do this since is a, generally quite compli- cated, function of . If ( ) 0 as we P 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
More Less

Related notes for STAT312

Log In


OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit