$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
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Taylor's Theorem

Let $U$ be an open and convex set and let $f\in C^{p+1}(U, \mathbb{R})$ for some integer $p\geq 0$. Suppose $\vec{a}, \vec{x}\in U$ and define $\vec{h}:=\vec{x}-\vec{a}$. Then, there exists some $\xi \in (0, 1)$ such that $$\begin{align*} f(\vec{x})&=f(\vec{a}+\vec{h})\\ &=f(\vec{a})+\sum_{k=1}^p \frac{1}{k!}\left[ (\vec{h}\cdot \nabla)^k f \right](\vec{a})+R+p(\xi), \end{align*}$$ where $R_p(t):=\frac{1}{(p+1)!}$\left[(\vec{h}\cdot \nabla)^{(p+1)} f\right] (\vec{a}+t\vec{h})

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Taylor's Theorem

Let $U$ be an open and convex set and let $f\in C^{p+1}(U, \mathbb{R})$ for some integer $p\geq 0$. Suppose $\vec{a}, \vec{x}\in U$ and define $\vec{h}:=\vec{x}-\vec{a}$. Then, there exists some $\xi \in (0, 1)$ such that $$\begin{align*} f(\vec{x})&=f(\vec{a}+\vec{h})\\ &=f(\vec{a})+\sum_{k=1}^p \frac{1}{k!}\left[ (\vec{h}\cdot \nabla)^k f \right](\vec{a})+R+p(\xi), \end{align*}$$ where $R_p(t):=\frac{1}{(p+1)!}$\left[(\vec{h}\cdot \nabla)^{(p+1)} f\right] (\vec{a}+t\vec{h})

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Hypothesis

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Results

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Proof

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FullPage
result
concepts
hypothesis
implications
proof
FullPage
result
concepts
hypothesis
implications
proof

Taylor's Theorem

Let $U$ be an open and convex set and let $f\in C^{p+1}(U, \mathbb{R})$ for some integer $p\geq 0$. Suppose $\vec{a}, \vec{x}\in U$ and define $\vec{h}:=\vec{x}-\vec{a}$. Then, there exists some $\xi \in (0, 1)$ such that $$\begin{align*} f(\vec{x})&=f(\vec{a}+\vec{h})\\ &=f(\vec{a})+\sum_{k=1}^p \frac{1}{k!}\left[ (\vec{h}\cdot \nabla)^k f \right](\vec{a})+R+p(\xi), \end{align*}$$ where $R_p(t):=\frac{1}{(p+1)!}$\left[(\vec{h}\cdot \nabla)^{(p+1)} f\right] (\vec{a}+t\vec{h})

Concepts

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Hypothesis

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Results

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Proof

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Taylor's Theorem

Let $U$ be an open and convex set and let $f\in C^{p+1}(U, \mathbb{R})$ for some integer $p\geq 0$. Suppose $\vec{a}, \vec{x}\in U$ and define $\vec{h}:=\vec{x}-\vec{a}$. Then, there exists some $\xi \in (0, 1)$ such that $$\begin{align*} f(\vec{x})&=f(\vec{a}+\vec{h})\\ &=f(\vec{a})+\sum_{k=1}^p \frac{1}{k!}\left[ (\vec{h}\cdot \nabla)^k f \right](\vec{a})+R+p(\xi), \end{align*}$$ where $R_p(t):=\frac{1}{(p+1)!}$\left[(\vec{h}\cdot \nabla)^{(p+1)} f\right] (\vec{a}+t\vec{h})

Concepts

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Hypothesis

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Proof

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result
concepts
hypothesis
implications
proof