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Use multiplication of division of power series to find the first three nonzero terms in the Maclaurin series for each function.

$ y = e^{-x^2} \cos x $

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02:38

Wen Zheng

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Campbell University

Harvey Mudd College

Baylor 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.

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 multiplication or divi…

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Use multiplication of divi…

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02:30

So for this problem, we want to use the multiplication of division of power Siri's in order to find the first three, none zero terms in the MacLaurin series. So we're given the function y equals e to the negative X squared coastline X. So we already know that co sign X McLaurin Siri's were given it already in table one. If you want to refer to that and it's negative one to the N of X to the to end time over two n factorial. So with that in mind, we also have e to the negative X squared, Equalling the sum of an equal zero to infinity of negative X squared to the end over in factorial. So with that in mind, another way we could write this would be if we want to rewrite it. We could write it as, um, negative one to the end. Since that's often preferred notation Negative one to the end times X to the two n over n factorial. So that being said, we see that four terms of each will probably be enough. So the first four terms of this will be one minus x squared over two factorial put us back into the boards over four factorial minus X to the sixth over six sectorial. And that obviously keeps going on. But we don't need any more. And then this is gonna be one minus X squared over two factorial plus X to the fourth over two factorial. Um, actually, this will be just X squared, Um, minus X to the sixth over three factorial and then plus go on. So what we'll do is we'll multiply these terms. So what we're gonna end up getting as a result is this right here and then we're gonna multiply this Bye. This right here. So when we do that, what we're gonna end up getting as a result, is simplifying it further. We'll get about one minus X squared over to minus two X squared over two. So these will combine to give us a negative three x squared over two so we can do that right there, and then we'll get plus X to the fourth over 24 plus two x to the fourth over to turning this into 24 will make this into 24. So we'll end up getting 25 x to the fourth over 24 so those would be our first three non zero terms within the MacLaurin series for the specific function.

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