 In this video, we're gonna find a power series representation for the algebraic expression one over one minus x squared. Now, in this situation, we have a repeated factor in the denominator. So there's a linear factor one minus x, but it shows up twice. And it turns out you can connect this rational function with this rational function by taking derivatives. Notice if f of x is one over one minus x, then it's derivative by the usual rules here. I can actually use a sort of a chain rule thing if I think of this thing as one minus x to the negative one power. Then by the chain rule, this derivative will look like negative one times one minus x to the negative two times the inner derivative, which is negative one. And so we get a one over one minus x quantity squared. So this builds the connection that we were trying to say here that if we take this function here to be f of x, this function right here is just f prime of x. Why is that significant? Well, the function f, we actually have a power series representation for it, right? So f of x, which looks like one over one minus x, its power series representation is the standard geometric series. The sum where n goes from zero to infinity of x to the end. And in expanded form, this will be one plus x plus x squared plus x cubed, et cetera. So this is a power series representation for f of x. If I wanna find a power series representation for f prime, that is if I wanna take the derivative of the function, I can just take the derivative of the power series, which we see before. You'll take the derivative of one, which is constant, it'll just go to zero. Derivative of x is a one, we get a two x, we'll get a three x squared, we'll get a four x cubed, just the use of the power right here. And so in general form, using the general form here, we're gonna get that the power series representation is gonna be the sum where n goes from zero to infinity. And then we're gonna get n plus one times x to the end, like so. And so if we write this power series represent, if this power series in a general form, we get something like the following. Now admittedly, you could have taken a slightly different approach. We could start this thing at one and go towards infinity, in which case you would get n times x to the n minus one. That's perfectly fine. That looks more like the power rule we get from derivatives. But in order to make this thing start at zero, we wanna shift everything down to zero. And so what that does is if you shift the starting point down by one, all of the n's that you're gonna see in the formula are gonna get shifted up by once. This is sort of like the standard thing we see with function shifts. You know, you have to turn right to go left, like Lightning McQueen knows very well. And so we get the following. And so now we have a power series representation of our function, one over one minus x quantity squared. And we obtain this by taking the derivative. So if we can find, that's to say if the function we're trying to represent as a power series, if it's the derivative of a known power series representation, we can take the derivative to work from there. And I should also mention that the original power series had a radius of convergence of one, and therefore its derivative will have that same radius of convergence. This power series representation is only valid when x is between negative one and one.