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The Lagrange and Cauchy forms of Taylor's remainder

Statement

Under the hypotheses of Taylor's Schlömilch–Roche remainder formula, there are points ξL,ξC\xi_L,\xi_C between aa and xx such that Rn,af(x)=f(n+1)(ξL)ι((n+1)!)(x−a)n+1R_{n,a}f(x)=\frac{f^{(n+1)}(\xi_L)}{\iota((n+1)!)}(x-a)^{n+1} and Rn,af(x)=f(n+1)(ξC)ι(n!)(x−ξC)n(x−a).R_{n,a}f(x)=\frac{f^{(n+1)}(\xi_C)}{\iota(n!)}(x-\xi_C)^n(x-a).

Facts & Assumptions

Given: The hypotheses of the Schlömilch-Roche theorem.

[L1]

For each natural 1≤p≤n+11\le p\le n+1, the Schlömilch-Roche theorem gives a point ξ\xi strictly between aa and xx such that Rn,af(x)=f(n+1)(ξ)ι(p)ι(n!)(x−ξ)n+1−p(x−a)p.R_{n,a}f(x)=\frac{f^{(n+1)}(\xi)}{\iota(p)\iota(n!)}(x-\xi)^{n+1-p}(x-a)^p. (Taylor's Schlömilch–Roche remainder formula).

[L2]

Factorials and natural powers obey The factorial n!n! and the falling factorial nk‾n^{\underline{k}}, defined by recursion in N\mathbb{N} and Integer powers ama^m, while the canonical embedding preserves products and positive naturals are nonzero (The canonical natural ι(n)=n⋅1F\iota(n) = n \cdot 1_F of a field, Canonical naturals are positive and strictly increasing).

Proof

technique · direct
1.1

Set p=n+1p=n+1 in [L1]. Then (x−ξ)0=1(x-\xi)^0=1 and ι(n+1)ι(n!)=ι((n+1)!)\iota(n+1)\iota(n!)=\iota((n+1)!), giving the Lagrange form.

L1L2algebra
1.2

Set p=1p=1. Since ι(1)=1\iota(1)=1, the formula becomes the Cauchy form.

L1L2algebra
2.1

These are the asserted special cases.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · next 3 levels

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Sources