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Suppose that $f(x)$ is differentiable on an interval centered at $x=a$ and that $g(x)=b_{0}+b_{1}(x-a)+\cdots+b_{n}(x-a)^{n}$ is a polynomial of degree $n$ with constant coefficients $b_{0} \ldots \ldots$ Let $E(x)=$ $f(x)-g(x) .$ Show that if we impose on $g$ the conditionsi) $E(a)=0$ii) $\lim _{x \rightarrow a} \frac{E(x)}{(x-a)^{n}}=0$then$$\begin{aligned}g(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime \prime}(a)}{2 !} &(x-a)^{2}+\cdots \\&+\frac{f^{(n)}(a)}{n !}(x-a)^{n}\end{aligned}$$Thus, the Taylor polynomial $P_{n}(x)$ is the only polynomial of degree less than or equal to $n$ whose error is both zero at $x=a$ and negligible when compared with $(x-a)^{n}$

$$g(x)=f(a)+f^{\prime}(a)+\frac{f^{\prime \prime}(a)}{2 !}+\frac{f^{\prime \prime \prime}(a)}{3 !}+\cdots+\frac{f^{n}}{n !}$$

Calculus 2 / BC

Chapter 10

Infinite Sequences and Series

Section 8

Taylor and Maclaurin Series

Series

Missouri State University

Harvey Mudd College

Idaho State University

Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

14:11

In mathematics, the partial sums of a series are the sums of all terms of the series except possibly the first and last.

06:48

Approximation properties o…

01:06

Let $a_{1}, \ldots, a_{n+1…

02:28

Let $f(x)=g^{\prime}(x) \f…

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