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Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function.$$ f(x) = x \cos 2 x $$

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$=\sum_{n=0}^{\infty} \frac{2^{2 n} x^{2 n+1} \cdot(-1)^{n}}{(2 n) !}$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Missouri State University

Oregon State University

Harvey Mudd College

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.

02:29

Use a Maclaurin series in …

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01:04

using the glorious years and table. Want to pretend that I'm serious? Okay. Showing us first you see off Michael groceries for two thousand. Next, which is because to one month have X squared plus x two four Book for Torrente pictorial minus extra six over six. Victoria. And so are so we're gonna put in this from there into here. I'm gonna get one witness, two x square over tutorial plus to ask to the core core. Booked her view, and this is equals to a signal. So case from zero to very and to cave tutorial two x to the power of two K times neck one to the horror of it. Okay, So in case you're oh, Touquet is just your and your terrorist one and we could take the first field terms. And if Keiko's one this is two extra Teo Teo twelve square over factorial times that one yet that's cracked, and we can actually insert acts in it. So the final result is gonna be too too too too Okay. Extruded to keep us one. Tom's not if you want to Pork. Okay. Over to King Pictorial.

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