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Find the Taylor polynomials $ T_3(x) $ for the function $ f $ centered at the number $ a $ Graph $ f $ and $ T_3 $ on the same screen.

$ f(x) = \ln x, $ $ a = 1 $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 11

Applications of Taylor Polynomials

Sequences

Series

Missouri State University

Campbell University

Oregon State University

Boston College

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.

02:28

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

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Find the Taylor polynomial…

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Find the Taylor polynomi…

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Find the Taylor polynomia…

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in this problem, whereas to expend nation Le Goff eggs using tethers full meal off order three on dhe around the given points or one a sequel to one. As you can see, we need to let the function at the given point. And 1st 2nd and third derivative Celeste Chocolate. First word 1st 2nd and third order Derivatives First knowing that two functions then she'll look effects first sort of derivative is one over X. From this, we calculate the second order derivatives as negative one over X cried and third order there too. Oh, thanks as to or X Cube. So now let's just use this expression and find turd Order for expression of dysfunction. He have a natural log off one plus one over one caps X minus one plus negative one over one times X minus one. Spread over two factorial. Plus we have to over one times X minus. Warm, cute or three Factorial. Do you know that natural ago form as zero. From this, we find court order approximation to dysfunction as X minus one minus X minus one. Scribed over to factorial plus X minus. Phone cubed over feet. How did I get that tree. Well, I know that three factories Eagle 26 on diesel can slot and we'll end up with three in the denominator. All right, You know the expression. Now, where s a graft to function and this approximation on the same graph? Um, we know that this is how the national lug bikes behaves. This is one and the approximation church order approximation behaves as such. Well, I didn't like this part. That me drove it again. All right? Okay. It's better now, is it, Jesse? Since this given function is, um, fairly a simple function, even the third order approximation does a pretty good job. And it captures pretty much the behavior in a way, you can see pretty large interval.

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