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    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?
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    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).
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    (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.
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    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
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    (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.
 
 
 
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