**Power Series Operations — Transcript & Summary | SozAI**
Source: https://sozai.app/transcript/power-series-operations/

Learn how to perform key operations on power series including substitution, exponentiation, and addition to simplify and find series representations.

## Key Takeaways

- Substituting x with kx scales each term by k^n in a power series.
- Replacing x with x^p raises the power of x in each term to p\*n.
- Adding two power series involves combining coefficients of like powers of x.
- Partial fraction decomposition can simplify complex rational functions into sums of geometric series.
- These operations enable expressing complex functions as power series for easier analysis.

## What the video covers

- Introduction to operations on generic power series F and G with coefficients a\_n and b\_n.
- Operation 1: Substituting x with kx in a power series results in multiplying each term by k^n.
- Operation 2: Replacing x with x^p changes the power series by raising x to the power p\*n.
- Operation 3: Adding two power series term-by-term by combining like terms with the same powers of x.
- Explanation of practical uses of these operations to find power series for complex functions.
- Example 1: Finding the power series for 2/(1 - x^3) by recognizing it as a geometric series with x replaced by x^3.
- Example 2: Using partial fraction decomposition to rewrite complex rational functions as sums of simpler geometric series.
- Demonstration of combining power series after decomposition to form a single power series representation.
- Emphasis on the utility of these operations for expressing more general functions as power series.
- Summary encouraging appreciation of these power series operations for broader mathematical applications.

Answers

## Questions about this video

What happens when you replace x with kx in a power series?

Replacing x with kx in a power series multiplies each term by k raised to the power of n, effectively scaling the series terms.

How do you add two power series together?

To add two power series, you combine like terms with the same powers of x by adding their coefficients, resulting in a new power series.

Why use partial fraction decomposition in power series?

Partial fraction decomposition breaks complex rational functions into simpler fractions, each of which can be expressed as a geometric power series, making it easier to find the overall power series representation.

## Full Transcript — Download SRT & Markdown

00:00

Speaker A

In this video, we're going to take a look at a few different operations that we can do with power series. So let's say we have two generic power series, one called F, one called G. F is the sum of a sub n x to the n. G is the sum of b sub n x to the n. So these can be different power series because they have different coefficients here. Alright, so the first operation is what happens if we take the x out of our power series and we swap it with a k times x. What changes will happen? Well, just take that x out on the right-hand side and switch it with k times x. K times x all to the n would be k to the n times x to the n. So this is how you do this operation. The second one, what if you take the x out again and this time, rather than a multiple of x, you switch it with a power of x, x squared or x cubed or something like that? Well, again, switch this out on the right-hand side, the x value with x to the p, and think about what happens. You'd have x to a power to the nth power, and in our algebra classes, we learned a power to a power, we multiply the exponents. So this will be the sum a sub n times x raised to the p times n. And the last and final property is what happens if you have one power series added to another power series. Well, as long as the index values start at the same place and they have the same variable terms x to the n, then as you're adding these terms, you'll see every term that an F matches a term with G. In other words, you have lots of like terms. So if you're adding these guys together, which are themselves summations, then you can take the like terms for each and combine them together. For example, the x squared term with the x squared term, the x cubed term with the x cubed term, and since you have x squared with x squared and x cubed with x cubed and so on and so forth for both of these terms, then you'll have a common factor of x to the nth power that you can factor out and really just add the coefficients together. Alright, so that's how you do these three different operations here. Now, I know what you may be thinking. You might be asking, okay, why do we care? Like, what on earth could these possibly be used for? Well, we actually have a lot of good uses for these guys. So let me just give you a couple quick examples. Let's say somebody asked us to find the power series for 2 over 1 minus x to the third, and you look at that, you say, well, if that had been 2 over 1 minus x, that would look like a geometric series, but it's a little weird having the x cubed in there. And in fact, I think I did a video on this earlier where we wrote this as what we called a geometric power series, and we did 2 over 1 minus x, and we wrote the summation as 2 times x to the n. Well, notice this is the same function as this with one difference. This isn't f evaluated at x, this is f evaluated at x cubed, right? Because you notice instead of an x, I've got an x to the third where x used to be. So how do we do this knowing that this is the power series representation of this function? Well, we're going to use this as a model, and we do the following. We would write this as the sum from, this was from n equals 0 to infinity of a times r to the n, but this would be x to the third to the n, which would make this the sum n equals 0 to infinity of 2 times x to the 3n, right? Which is exactly what this property back here said. This was actually property number 2 that we're using right here. So these properties here can really help when you try to find the power series for certain functions. Here's another one. Here's our last one. This is what we'll close with. Let's say somebody gave us this function here and they asked us for its power series representation. We have no idea. I mean, this looks nothing like a geometric power series. It looks like no power series that I'm aware of. But you might notice something. You this guy, and if you noticed that this denominator right here factored as 1 minus x times 1 plus x, hopefully you would realize that you can do partial fraction decomposition on this guy and actually decompose it into a couple of different fractions, a couple of different rational expressions. So here we go. I went ahead and did this. Notice that this guy would be equal to 2 over 1 minus x minus 3 over 1 plus x. Now, how did I get that? Well, on some scratch paper, I did a partial fraction decomposition and I decomposed it into these two smaller pieces. Now, why is that a good idea? Well, let me just ask you, could you find his power series? Does that look like maybe a geometric power series a over 1 minus r? Yes, I think we can. And could we find his geometric power series? Might take a little work, but I think we can do that as well. Okay, so this guy's power series is a times r to the n, and this one has a little bit more algebra to it than I think I'll discuss in this video. Basically, in short, you have to account for this plus sign. It should be a minus sign, so we would consider this as like minus negative x, which is going to change things a little bit in the geometric power series. But in effect, we still have a times r to the n, but that negative x to the n separates into negative 1 to the n times x to the n. So that's a lot of details I'm not going to go through in particular. That's not really the point of this video. My point is simply that we can write each of these as a power series. But now what do we have? We have the sum or difference of two different power series, and that property that we had back here said that if you have the sum of two power series, you can combine these into a single power series. Alright, so what we can do here is we can write one giant series, and we'll have this common term in this common term being subtracted, and since they both have an x to the nth power, we can pull that x to the nth power. So this guy's power series representation is the very last line, and the only way we could have known that is if we done partial fractions and then used that property that we just learned here in this video. So hopefully you do appreciate some of these operations that we can do on power series, and hopefully they'll let you write more general functions as power series using those three operations.

00:12

Speaker A

sub NX to the N G is the sum of B sub NX to the N so these can be different power series because they have different coefficients here alright so the first operation is what what happens if we

00:24

Speaker A

take the X out of our power series and we swap it with a K times X what what changes will happen well just take that X out on the right hand side and switch it within K times X K times X all to the

00:39

Speaker A

N would be K to the N times X to the N so this is how you do this operation the second one what if you take the X out again and this time rather than a multiple of X you you switch it with a

00:53

Speaker A

power of X x squared or X cubed or something like that well again switch this out on the right hand side the x value with X to the P and think about what happens you'd have X to a power to the nth power and in our

01:09

Speaker A

algebra classes we learned a power to a power we multiply the exponents so this will be the sum a sub n times X raised to the P times in and the last and final property is what happens if you have one

01:25

Speaker A

power series added to another power series well as long as the index values start at the same place and they have the same variable terms X to the N then as you're adding these terms you'll see every term that an F matches a term with

01:42

Speaker A

G in other words you have lots of like terms so if you're adding these guys together which them which are themselves summations then you can take the like terms for each and combine them together for example the x squared term with the

01:56

Speaker A

x squared term the X cubed term with the X cubed term and since you have x squared with x squared and X cubed with X cubed and so on and so forth for both of these terms then you'll have a common

02:08

Speaker A

factor of X to the nth power that you can factor out and really just add the coffee together alright so that's how you do these three different operations here now I know it I know what you may be

02:19

Speaker A

thinking you might you know might be asking what okay why do we care like what what on earth could these possibly be used for well we actually have a lot of good uses for these guys so let me

02:29

Speaker A

just give you a couple quick examples let's say somebody asked us to find the power series for 2 over 1 minus X to the 3rd and you look at that you say well if that had been 2 over 1 minus X that

02:43

Speaker A

would look like a geometric series but it's a little weird having the X cubed in there and in fact I think I did a video on this earlier where we wrote this as a what we called a geometric

02:53

Speaker A

power series and we did 2 over 1 minus X and we wrote the summation as 2 times X to the N well notice this is the same function as this with 1 difference this isn't f evaluated at X this is f

03:10

Speaker A

evaluated at X cubed right because you notice instead of an X I've got an X to the third where X used to be so how do we do this knowing that this is the power series representation of this

03:23

Speaker A

function well we're going to use this as a model and we do the following we would write this as the sum from this was from N equals 0 to infinity of a times R to the n but this would be X to the 3rd to

03:43

Speaker A

the N which would make this the sum N equals 0 to infinity of 2 times X to the 3 n right which is exactly what this property back here said this was actually the property number 2 that we're using right here so these

04:01

Speaker A

properties here can can really help when you try to find the power series for certain functions here's another one here's our last one this is what we'll close with let's say somebody gave us this function here and they asked us for

04:14

Speaker A

its power series representation we have no idea I mean this looks nothing like a geometric power series it looks like no power series that I'm aware of but you might notice something you this guy and if you noticed that this

04:31

Speaker A

denominator right here factored as 1 minus x times 1 plus X hopefully you would realize that you can do partial fraction decomposition on this guy and actually decompose it into a couple of different fractions couple different rational expressions so here we go

04:49

Speaker A

I went ahead and did this notice that this guy would be equal to 2 over 1 minus X minus 3 over 1 plus X now how did I get that well on some scratch paper I did a partial fraction

05:02

Speaker A

decomposition and I decomposed it into these two smaller pieces now what why is that a good idea well let me just ask you could you find his power series does that look like maybe a geometric power series a over 1

05:15

Speaker A

minus R yes I think we can and could we find his geometric power series might take a little work but I think we can do that as well ok so this guy's power series is a times R to the N and this one has a little bit

05:31

Speaker A

more algebra to it than I think I'll discuss in this video basically in short you have to account for this plus sign it should be a minus sign so we would consider this as like minus negative X which is going to change things a little

05:45

Speaker A

bit in the geometric power series but in effect we still have a a times R to the n but that negative X to the N separates into negative 1 to the N times X to the N so that's a lot of details I'm not

06:00

Speaker A

going to go through in particular that's not really the point of this video my point is simply that we can write each of these as a power series but now what do we have we have the sum or difference

06:11

Speaker A

of two different power series and that property that we had back here said that if you have the sum of two power series you can combine these into a single power series all right so what we can do

06:24

Speaker A

here is we can write one giant series and we'll have this common term in this common term being subtracted and since they both have an X to the nth power we can pull that X to the 8th power so this

06:37

Speaker A

guy's power series representation is the very last line and the only way we could have known that is if we done partial fractions and then use that property that we just learned here in this video so hopefully you do

06:51

Speaker A

appreciate some of these operations that we can do on power series and hopefully they'll let you write more general functions as power series using those three operations

Topics: power series power series operations geometric series partial fraction decomposition series substitution series addition mathematics calculus algebra Firefly Lectures

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