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Use the definition of a Taylor series to find the first four nonzero terms of the series for $ f(x) $ centered at the given value of $ a. $

$ f(x) = \frac {1}{1 + x}, $ $ a = 2 $ $

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First four non-zero terms of the Taylor series are:$\frac{1}{3},-\frac{1}{9}(x-2), \frac{1}{27}(x-2)^{2},-\frac{1}{81}(x-2)^{3}$

02:34

Wen Zheng

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Campbell University

Baylor University

University of Nottingham

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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Use the definition of a Ta…

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a. Use the definition of a…

02:29

So we want to determine the first four non zero terms of the tailor Siri's, which we ultimately know that but we'll have as a result, is the first term of our Taylor Siri's. We know that a tailor Siri's, is F to then on minus one times to I'm X minus two m s f to the and minus one. Crime of to I am inspects minus two to the end, minus one all over are going to be over and minus one factorial. So, with this in mind, um, what we'll do is plug in different values. So the first term that we have, I will be a zero. So overall one here. So we just have f of to, um f of two X minus 20 So it's just going to give us one so f of to over zero factorial. We know that zero factorial is just one and f of to is one third. So we get one third I'm to or one third, um, over one, which is just going to give us one third. And for the second term, what we'll have is we know that f prime of to since that will have or give us a negative 1/9 IMEs. In this case, we have X minus two. So X minus two to the one. Yeah, and this will be just one factorial down here, so I end up getting as a result is going to be equal. Thio a negative 1/9 times X minus two. And for our third term of the tailor Siri's. What we have is this will be f double crime until which we know to be to over 27. Then times X minus two squared. We know that this will be two factorial because it's three minus one so two factorial and that's going to end up giving us one over seven because the two factorial is just 2 1/27 times X minus two spread. Then lastly, we have our final Taylor. Siri's a term. This will be asked triple prime few, which we know is a negative to over 27. You know, this will be three factorial. This will be X minus two cubed three. Factorial is six. So what we'll end up getting as a result is going to be negative. 1/81 x minus two. Cute. I've just come from simplifying this portion right here. Six divided by that right there. So this will become our final answer that we have for the third term. And these are other subsequent terms that we have.

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