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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) = \sinh 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

Campbell University

Baylor University

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.

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Find the Maclaurin series …

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I find the men lanaseries for f of x using the definition all right, so we're going to use the definition to find the lenses for f of x so evidently goes to n from 0 to infinity derivative at 0 over in factorial and x power. So f x is sine x and prime is cosine x, point again after prime equals sine x and so on. So it's like an iteration and we know that f. 0 is 0, and i prize is this 1 after prime and i've triterite prime at 0 is still 1. So, to get things clear, we can write down the definition of sine x and cosine x, so we're plugging all this data into or equation 1 and we're going to have a that's going to be so inform okay, so its x plus x, cubed over 3 factorial Plus 55 or 5 factorial and plus so all the even number, all the odd terms, i'm sorry and the final answer is going to be so k from 1 to infinity x to the power of 2 minus 1 to minus 1 factorial set. So that's all the odd terms are terms for the power of x and the reader. Convergence is going to be the whole real line. So r, o c is just the whole real line. Here'S infinity.

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