Hey there! Sign in to join this conversationNew here? Join for free
    • Community Assistant
    • Thread Starter
    Offline

    18
    ReputationRep:
    For an (n+1) times differentiable function ;

    \displaystyle f(x) = f(x_0) + (x-x_0)f'(x_0) + \frac{(x-x_0)}{2!}f''(x_0) + \cdots + \frac{(x-x_0)}{n!}f^{(n)}(x_0) + E_n(x)

    Where \displaystyle E_n(x) = \frac{(x-x_0)^{n+1}}{(n+1)!}f^{(n+1)}{( \xi)} for some \displaystyle \xi \in (x_0,x)

    What is the reason for the last term, mainly why does \xi have to be in the open interval?
    Offline

    17
    ReputationRep:
    Well because the existence of such a \xi follows from Rolle's Theorem on the interval [x_0,x] (this will be clear in the proof of this version of Taylor's Theorem).
    • Community Assistant
    • Thread Starter
    Offline

    18
    ReputationRep:
    (Original post by IrrationalRoot)
    Well because the existence of such a \xi follows from Rolle's Theorem on the interval [x_0,x] (this will be clear in the proof of this version of Taylor's Theorem).
    Okay, this is actually stated, I'm just a bit tired, but thank you.
    Offline

    10
    ReputationRep:
    This is the Lagrange Remainder term. I can not tell you much about it but I know it depends on the Mean Value Theroem (hence  \zeta \in (x_{0}, x) ) but the proof of it can be searched, such as this: http://mathonline.wikidot.com/taylor...ange-remainder
    • Community Assistant
    • Thread Starter
    Offline

    18
    ReputationRep:
    (Original post by simon0)
    This is the Lagrange Remainder term. I can not tell you much but the proof of it can be searched, such as this: http://mathonline.wikidot.com/taylor...ange-remainder
    Thank you.
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    Did TEF Bronze Award affect your UCAS choices?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.