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Find the Maclaurin series for $ f(x) $ using the definition of a Maclaurin series. [ Assume that $ f $ has a power series expansion. Do not show that $ R_n (x) \to 0. $] Also find the associated radius of convergence.

$ f(x) = \cos x $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Missouri State University

University of Nottingham

Idaho State 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.

05:25

Find the Maclaurin series …

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okay. So fundamental or a serious forever Backs used death and nation that McLaren Siri's assume that has the power. She respond to notion that not show that are still in the backs Goes to zero. Also found associated readers of convergence. And here we have at Max equals Coulson necks. Okay, Sylvan. The first fine cousin zero, which is one. And the derivative is money. Sanks as your zero, and it's gonna be minus one. So we're gonna fund that. Here's the pattern of the George have at zero so foreign before insecurity. And in this and is a natural number is going to be one and foreign plus one or four plus three. This is zero four plus two is gonna be next one. All right, showing. And I use this to do, like, over MacLaurin series for FX silver by definition, this is thie instituted of F zero over a pictorial Times X to the power in. So this gonna be We're plugging this song so it's gonna be like X minus X square with you, Victoria. Plus, that's for over four Victorian minus extra six over Vittorio class and so on and so on. This is actually say case from zero to infinity that you wanted Power K and X to the poor of two K over to K. Victorio. So one case zero, This is positive. One and no, actually, this is all too k. Plus, I'm sorry. Here is one is not ass, though. Yes, this to Kay and Victorio. Yeah, this is over. Final results for the metal Orin Siri's of FX and the readers. Convergence is just the choral line.

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