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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) = \cosh x $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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.

04:32

Find the Maclaurin series …

01:50

02:14

07:00

05:25

10:06

02:46

03:29

02:00

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02:13

find my corn serious for half of ax using the definition of met Lauren Siri's. Okay, So gonna farm. We're going first. Find the primes. The rhythms for that's so zero at zero equals half of eat with zero. Plus, it was not a zero, which is zero, and it goes to one, and a crime at zero is gonna be That's six zero is going to zero and so on were going to find that so forty even order of the derivative at zero is gonna be zero. And for the old number for the other of us is going to one. So we plug in this data into over definition of McLaren, Siri's, and we're going to have This is our definition, and we know that for your r. O. C. It is the affinity because it converts for the whole realign end. With plugging this data, it becomes one purse. So for the even, Order of threw him is just a one and for all, Member, it's just a zero. So Okay, it wasn't this I got So which is equals to K equals two in King. Sorry, Kay. From zero to infinity, Extra pole of two in over two K to Victoria. Yeah, that's all the answer

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