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Express the function as the sum of a power series by first using partial fractions. Find the interval of convergence.$ f(x) = \frac {2x - 4}{x^2 - 4x + 3} $

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$(2 -x)\left(\frac{1}{3} \sum_{n=0}^{\infty} \frac{x^{n}}{3^{n}}+\sum_{n=0}^{\infty} x^{n}\right)$interval: (-1,1)

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 9

Representations of Functions as Power Series

Sequences

Series

Campbell University

Oregon State University

University of Michigan - Ann Arbor

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.

03:05

Express the function as th…

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04:50

0:00

13:23

$11-12$ Express the functi…

01:08

Find a power series repres…

02:15

02:47

03:25

09:44

02:22

12:47

01:18

Okay, So expect function as the some power Siri's first using partial corrections and find the interval convergence. Okay, so everyone x as two times X minus two. And the denominator is that's minus one times. It's my three. Okay? And this is just and that's money. Three. The inverse of that's my three minus one over X minus. What? So inside here, it's always this part. So this difference equals two X minus three times X minus one and the Obama's gonna be accidents one minus X men three. So this is to actually yes. So we have this equipments. Okay, So you continue doing this is going to transform too, which is pull out three. And the inside is gonna be a sword. Three months, one and manager minus X minus wesser. This is plus one over one month's X. And here is which is pull out next one. So, uh, this turns to be two months sex, and here it goes to one minus has all three. Yet and so this part, this kind should be so it is expand, then turn by term silly, becomes extra power inthe rated out and inform zero to in a Plus, that's power in from zero to infinity. So that is thie. Some of power, Siri's. And what is Thie? Inderal of convergence. So we require Iris. That's over three is less than one. The absolute alibis, Authoress. That's what ended the actual axe. Is this one. So that that becomes silly, actually. Well, if X is less than three and someone so Leslie Thean, welcome earns is gonna be once want to what?

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