Results 1 to 2 of 2

Math Help - integrals and taylor's remainder function

  1. #1
    Member
    Joined
    Sep 2009
    Posts
    104

    integrals and taylor's remainder function

    I am stumped as to how to approach this one:
    Prove that the remainder term in Taylor's theorem can be replaced by
    R_{n}(x)=\frac{1}{n!}\int^x_a f^{(n+1)}(t)(x-t)^ndt provided that f^{(n+1)} is Riemann integrable.

    [Taylors remainder term: R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    Mxico
    Posts
    721
    Use integration by parts with u'=f^{(n+1)}(t) and v=(x-t)^n and when done rememeber that R_{n-1,a}(f)-R_{n,a}(f)=P_{n,a}(f)-P_{n-1,a}(f)= \frac{ f^{(n)}(a)(x-a)^n}{n!}

    PS. Notice that behind all of this the argument is inductive so you should try to prove the case n=1,2 apart
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Taylor Polynomial and Remainder
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 10th 2010, 03:05 AM
  2. Taylor remainder
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 24th 2010, 01:00 PM
  3. Taylor's Theorem with Remainder
    Posted in the Calculus Forum
    Replies: 0
    Last Post: May 2nd 2009, 02:33 PM
  4. Taylor Polynomial & Remainder
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 7th 2008, 12:43 PM
  5. Taylor series and remainder
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 2nd 2008, 10:54 AM

Search Tags


/mathhelpforum @mathhelpforum