CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Multivariable Taylor formula with $o (∥ h ∥^{k})$ remainder

Statement

Let $k \in N$ with $k \geq 1$ , let $V \subseteq R^{m}$ be open and convex, let $a \in V$ , and let $f \in C^{k} (V)$ . Then, as $h \to 0$ with $a + h \in V$ ,

$f (a + h) = T_{k} f (a; h) + o (∥ h ∥^{k}) .$

Facts & Assumptions

Given: The hypotheses of the statement and small $h$ with $a + h \in V$ .

[L1]

By Multivariable Taylor formula with a Lagrange remainder along a line segment, applying the multivariable Lagrange formula with degree $k - 1$ gives some $θ_{h} \in (0, 1)$ such that

$f (a + h) = T_{k - 1} f (a; h) + \sum_{∣ α ∣ = k} \frac{D^{α} f (a + θ_{h} h)}{ι (α!)} h^{α}$

[L2]

For every $∣ α ∣ = k$ , $D^{α} f$ is continuous at $a$ ( $C^{k}$ maps and multi-index derivative notation in Euclidean space).

Proof

technique · direct

1.1

Subtract the degree- $k$ Taylor polynomial from the equality of [L1]. The remainder is the following.

L1algebra

$\sum_{∣ α ∣ = k} \frac{D^{α} f (a + θ_{h} h) - D^{α} f (a)}{ι (α!)} h^{α} .$

2.1

For $h \neq 0$ , divide the absolute value in step 1.1 by $∥ h ∥^{k}$ . Since $∣ h^{α} ∣ \leq ∥ h ∥^{k}$ , it is bounded by the finite sum of the coefficient differences divided by $ι (α!)$ . As $h \to 0$ , also $a + θ_{h} h \to a$ , so every term tends to zero by [L2].

step 1.1L2algebra

3.1

This proves that the remainder in step 1.1 is $o (∥ h ∥^{k})$ , and the subtracted polynomial is $T_{k} f (a; h)$ by The multivariable Taylor polynomial in multi-index notation.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

MAT237 notes: Taylor's theorem in several variables (standard reference, not scraped)

Multivariable Taylor formula with o(∥h∥k)o(∥h∥k) remainder