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Find a power series representation for the function and determine the radius of convergence.$ f(x) = \left( \frac {x}{2 - x} \right)^3 $

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$-\frac{x^{3}}{4}\left(\sum_{n=2}^{\infty} \frac{n \cdot(n-1) x^{n-2}}{2^{n}}\right)$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 9

Representations of Functions as Power Series

Sequences

Series

Missouri State University

Harvey Mudd College

Baylor University

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.

01:33

Find a power series repres…

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02:51

okay. Find the power. Serious limitations function and turning the readers of murders affects. It was That's cute. Comes one or two months X cure so we can use the trick. And we claimed that it was to execute times little one half times. The second one there were two minus X. So we're gonna fund it is true or not. So what is it? Good one? Or ask Teo to overtax whenever two mothers? Yes. So the first derivative of it becomes too when there are two months asked our two and not one. So it becomes too minus one or two months esque square and we can do this. Do it again. So the double direct vehicles the directive of this dysfunction it becomes, too. That derivative of minus one times two months. Extra power from one is two. So this is just two months Extra hour, one three times minus two terms ninety one. No, this is on. So this transferred out isn't one is two. One is to co over two months. Ex cute. Yes. And here we have number one half so naked one hat and later to the cancer dot with Siri's in a quote. This is It is true. Okay, so we're gonna continue doing this. We can read this part in things get easier to do because it's very easy to expand when you were two months X. Okay, so this is just to the cube. Next, ask Cube asked as Cuba or two times one half and second review of one minus half of ax Silla. It becomes too. Not you excuse or four times the second directive Extra power of end to tip our inn in from zero to infinity. And this is, um, begin minus X cube. Oh, four and weakens first through the first directive, Piso and from zero is showing family. But which out? So the first time it is extra power of zero over to deposit. So it is won and won Has the director zero. So now and should be former once evenly. So it becomes too X to power on minus one and toms and ends here. Yeah, otherwise we're going to find zero, so it doesn't matter, because if we put well, who would know it matters. So the sea if you put a little syrup in front to seventy. So there's a there's a mistake because X to the power zero minus one is is thie inverse of acts. So it's not true. It's not bad form for the power. Siri's. Yes. So that's why we should modify this in that from one community on DH. Yeah, that's sure, because when zero this term is a constant and now so the identity is thiss part and community doing the differentiation, it becomes two months of skew go for and here, hands front to Because we're in this one. The first part is the constant the first term. So can you bring that there, Load him for this part? It becomes then times in minus one Excellent, too powerful and managed to and over to the heart one to the power n So this is our final answer for the representation of power. Siri's. So this is the final answer. Yeah. Okay, so that's us to find the readers of convergence. So here we have the absolute off X over two is less than one implies I am still available and is less than two sword. So which means thie reduce of convergence is our equals two. Okay,

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