Approximation by Taylor Polynomials

Definition

Let $a < b$ be real numbers.

Let $f : \closedint a b \to \R$ be a real function.

Let $n \in \Z_{>0}$.

Let $\xi \in \closedint a b$.

Let $f$ be $n$ times differentiable at $\xi$, where all derivatives are one-sided if $\xi = a$ or $\xi = b$.

Let $T_n$ be the Taylor polynomial of degree $n$ for $f$ about $\xi$, that is:

$\ds \map {T_n} x = \sum_{i \mathop = 0}^n \frac {\map {f^{\paren i}} \xi} {i!} \paren {x - \xi}^i$

Then:

$\ds \lim_{x \mathop \to \xi} \frac {\map f x - \map {T_n} x} {\paren {x - \xi}^n} = 0$

where the limit is one-sided if $\xi = a$ or $\xi = b$.

Proof

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Sources

• 1981: Karl R. Stromberg: An Introduction to Classical Real Analysis: $4$.Differentiation