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Find the Taylor series for $ f(x) $ centered at the given value of $ a. $ [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) = 1/x,$ $ a = -3 $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Oregon State University

University of Michigan - Ann Arbor

University of Nottingham

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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.

02:34

Find the Taylor series for…

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08:22

The problem is finding the Taylor series for Alfa Back Center that has given value of A and finding associated readers of convergence. So first I have negative three is equal to negative 1/3. A promise is a cultural negative one over expire? Uh huh. From prom is equal to two over x cube and F third derivative. This is a cultural negative three factorial over access to the part of four so we can find that the Earth derivative is equal to negative one to the power of and and times minus one factorial over access to the power of em. Here's a plus one, then f n negative three. It's equally true. Next, the one to the power of end times and minus one factorial over negative three to the power of M s one which is equal to negative and minus one factorial over three to the power of plus one. So the Taylor series for half a wax is some from zero to infinity half and derivative at the point. Negative three over and factorial times X plus three to the power of end, which is equal to some from 0 to 20 Negative one over 10 times three to the power of plus one harms X plus three to the power of the pen. Have a compute limit and go straight. Infinity. Absolute value of make 21 over I'm plus one times ready to power of M plus two over negative one over N times three and plus one, which is equal to the limit, and Austria Final T and over three times and plus one, which is equal to 1/3. So the Raiders of Convergence is equal to 1/1 over three, which is equal to three.

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