Doubt regarding Taylor's theorem

1. Aug 5, 2014

utkarshakash

1. The problem statement, all variables and given/known data
In Taylor's series, the Lagrange form of remainder is given by
$R_n = \dfrac{f^{n+1}(t)(x-a)^{n+1}}{(n+1)!} \\ t \in [a,x]$

whereas my book states that it is given by
$\dfrac{h^n}{n!} f^n (a+ \theta h) \\ \theta(0,1)$

I don't see how these two are interrelated. Can anyone explain ?

2. Aug 5, 2014

Ray Vickson

How is $h$ defined in the expression from your book? Obviously, if $h = x-a$ then your book's expression is the same as the Lagrange version, because if $\theta \in [0,1]$ then $a + \theta h$ is a number in $[a,x]$.

3. Aug 5, 2014

utkarshakash

But the book's version has nth derivative. Shouldn't it be n+1?

4. Aug 5, 2014

Staff: Mentor

It would be helpful to include some of the accompanying text in your book. It's possible that your text is giving the remainder after the terms of degree n - 1, while the more usual form you also show gives the remainder after terms of degree n.

5. Aug 5, 2014

Ray Vickson

I agree with Mark44. One form gives the remainder after an (n+1)-term polynomial (where the powers of (x-a) go from 0 to n) while the other (perhaps) gives the remainder after an n-term polynomial (where the powers go from 0 to n-1).