Turn on thread page Beta
    • Community Assistant
    • Thread Starter
    Offline

    18
    ReputationRep:
    Community Assistant
    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:
    Community Assistant
    (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

    11
    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:
    Community Assistant
    (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.
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: November 2, 2017

University open days

  1. University of Bradford
    University-wide Postgraduate
    Wed, 25 Jul '18
  2. University of Buckingham
    Psychology Taster Tutorial Undergraduate
    Wed, 25 Jul '18
  3. Bournemouth University
    Clearing Campus Visit Undergraduate
    Wed, 1 Aug '18
Poll
How are you feeling in the run-up to Results Day 2018?
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

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

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