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(a) Use differentiation to find a power series representation for $ f(x) = \frac {1}{(1 + x)^2} $What is the radius of convergence?(b) Use part (a) to find a power series for $ f(x) = \frac {1}{(1 + x)^3} $(c) Use part (b) to find a power series for $ f(x) = \frac {x^2}{(1 + x)^3} $

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a. $(-1)^{n+2}=(-1)^{n}$b. $R=1$c. $R=1$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 9

Representations of Functions as Power Series

Sequences

Series

Campbell University

Oregon State University

Harvey Mudd College

Baylor 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.

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(a) Use differentiation to…

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(a) Find a power series re…

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$$\begin{array}{l}{\text {…

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(a) Use a geometric series…

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Find a power series repres…

Okay, So first used depreciation to fund a power source reputation for FX part, eh? So we're gonna find effects such that the F over the X equals lower case FX, and it's very easy to find it. So Kepco FX is just one word. One has X, of course, is nectar. So let's hang out so that the real bitch of it is gonna be minus one over one plus x square times minus what? So this is just whenever else because one to the power of two. Yes. So we know that Keppel FX is very easy to be a factor out to be, to expand it. So this is just minus one times, man. It's wanted power in extra power and from zero to infinity. And it was in this squid. Then I with zero months with powerful plus one from sex to powerful men. Okay, silicon, a different shade. This cab, let's turn my turn from this summation and, uh, lower his seven equals the derivative of Canada. FX equals. So to match her, you're gonna find minus twenty Power plus one and X to the power of in minus one. And actually, this is Twitter or ICO two one. Because when in the zero, it becomes a constant And for any counsel we know interruptus goes to zero. So it has changed the index. It goes from one and over. The answer for Paris is lower. His f X equals so minus one to power and class one times into power today, then times X to the power n minus one. Okay. All right. And if the readers of converters So what if the readers of converters we just noticed the time here. So that is where we just expanded a serious. So it requires minus X, the model man just less than one. So that means there is commitments is gonna be absolute. Bella backs is less than one. Okay, so we just raised our this part and keep the results two, figure out the party and see we're second. All right. So poor be issues. Party to find the power serious for sequels when you are wood plus us. The cube. Okay, so this seems very similar. We used the similar methodology, So if you're gonna rewrite the power serious reputation for part A So that isthe and it's weird. And I called to one minus. Wanted horse on plus one and times X to the power in minus one. Okay. And Ah, so what is Thie? Entire derivative of one hour? What? Plus X to the power of three. It's very easy to fund it. This Mary. So you're in the world this with this is just Ah, when our one plus x square and here is minus one third over here, I think. No, it's what? Half minus one. Okay, let's take it out. So minus one or two times one plus square, that is just one half times went past next with horror minus two. So the road a little bit because one plus x to the power from one three. Yes. Oh, we got this hunted eroded and held us listening to FX. So this was just one is one half times at Nick's. So what does this imply? This implies that. So let's say this is one of us, and this is up to us, and this is three X, and we know that up to us, it's just it's just one once one half times the derivative. Thanks yet. So how do we expand to FX? We just used the same methodology. We're gonna differentiate half of one next time, but her. So it goes one or two months one, too. And we're gonna find it The radio off this part. So again, when any was one service term is a constant. So when now Ripper and their squeezer, then Michael, too. And that the river there that is going to be in terms in minus one times X power and money. Stupid. So that's it. This part, It's the answer for Poppy. Yeah, And first you write this, we're going to move this results to here and do three. So Katherine is again. We can use the result for Poppy to fund Guthrie, and that's rather easy. I think we do not need to use the derivative unsteady with that makes we just do. It's by. So this is the answer for party after us equals this one. And for part three, we know this. That's three ex Pecos at square tons to X, so we can just replenish write down the answer directly. So there's going to be next you one half times into the power off now, and it's greater than equal to two I once went off in plus one in the most one, then in X to the power of on minus two plus two. So that is just extra power in. So the answer's no path. Ray's here. That's the power. Serious. Reputational par three for F of three of three of my ex. Yeah, and now this over. Final answer. Don't forget the rays of convergence here is minus ones. Who What? Okay.

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