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Let $f(x)$ have derivatives through order $n$ at $x=a$. Show that the Taylor polynomial of order $n$ and its first $n$ derivatives have the same values that $f$ and its first $n$ derivatives have at $x=a$

$$P_{n}^{\prime}(a)=f^{\prime}(a), P_{n}^{\prime \prime}(a)=f^{\prime \prime}(a) \text { and } P_{n}^{n}(a)=f^{n}(a)$$

Calculus 2 / BC

Chapter 10

Infinite Sequences and Series

Section 8

Taylor and Maclaurin Series

Series

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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.

01:55

Let $f(x)$ have derivative…

01:34

Let$$f(x)=\left\{\begi…

43 for this problem. Just right down with the territories off order and news. And you can find out the answer So that that BP here next This he goes to Sigma k found year old. Uh, Okay, Max. Ready by Capt. Auriol, Times X minus a. Okay, so just take. So it's ah polynomial degree and right, so just take a prime. You see, gusto is constant, so Okay. From one toe. Hey, no times X minus a the u K. Minus one. Right. Here's a k times when I work a tutorial. So it's just k minus one. Victoria. So that's that's it. So ps cry, eh? It's just seacoast two in case one. So the case one that's of Prime lacks a prime a right, And when case, too. So that's a zero. Okay. Quit heard into. And that zero gain. So So the prime off. He as acts, it's just a prime, a right, And just by induction, you can find out that p prime crime. And hey, it's just people who have private, eh? So that's the same just by induction. Just take the director with one times and again and again you can find out

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