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Find a power series representation for the function and determine the radius of convergence.$ f(x) = \ln (5 - x) $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 9

Representations of Functions as Power Series

Sequences

Series

Campbell University

Oregon State 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.

04:03

Find a power-series repres…

01:33

Find a power series repres…

05:55

10:16

$15-20$ Find a power serie…

05:30

05:50

fun power, serious reputation for function F ax and determine the readers of convergence. Okay, so I'm gonna transport axe into lawn with this pool of five, and it's because against one ministers or five And what this This is long fine, plus lawn one minus X or if I and we're going to spend this part, actually, what is that? So, uh, this is just the integral of when there were one minute six or five the X. So let's have not. Maybe we should modify this by some constant K here, so let's try this. So, what is the Negro Levitt? Is this on long one minus plus to power? That's over five. And I think there is some constant here, so Okay, let's do this. So, what is the relative of long one minus? That's or fight. So this is one hour four, five times, not people one thing. Okay, so we confirmed that the way it goes, Nothing. One fifth. So, uh, yeah, okay. And actually, we could spend this part. So one of our women, this X for fighting is very easy to be expanded into hair Siri's. So this particle becomes to end from zero could be and, uh, extra power in your health. And here I have the differential of reader, the X. So we're gonna integrate this one through the term and actually were able to change the other integral. And the summation in there on his son. So on with the first girl extra card and ORF of love in the X and this summit. And this is just equal to our own life. By my eyes, when you were to the power from past one and some notation Inigo is explored in the eggs. It's rather familiar to us. So on. Okay, let's first you write this part and this is just lawn five minus when there are servants, US one and the integral is gonna be extra power and plus one over and us one. So that's the girl. Yes. Okay, so now we have our results. That's the power of serious reputation for the function for FX on determine the readers of Convergence. So how to find the richest convergence? We gotto push up on it from the step where we just expand dysfunction. So it's this equation we require that that's or fine is from zero ice from the ones who want question, place, axis from knocking fight to five. So that's the rays of convergence, okay?

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