[MUSIC] So how to divide from a power series. It turns out that in many cases, it's possible, but not always. So, suppose we have a power series A(q) = a nought + a1q + a2q squared +. Etc. We say that the power series b is the inverse of A(q), if their product is equal to 1. So, definition B(q), which will be denoted by A(q) inverse, is the inverse. Of A, if their product is = 1, so if. B(q).A(q) = 1. Example. Let's take the geometric progression. I claim that its inverse. Is 1-q. So as I said, well this is a polynomial and q, but a polynomial is also a formal power series. So let us multiply them, and let us see that this is indeed an inverse to this guy. Indeed (1 + q + q squared +...)(1- q) = which has expand this product, 1 + q + q squared + etc- q- q squared- q to the 3rd -..., and you see that all terms in this sum, except of the first one, except for 1, they vanish. So this is just 1. So. The inverse to the geometric progression is 1- q, and this is exactly what we meant by saying that, (1 + q + q squared +...) = 1/1-q. So,the product of this thing and the denominator is 1. And so this power series is invertible. Our next claim is that, a power series is invertible if and only if, its lowest term is not 0. Theorem. A(q) has an inverse. If and only if. A naught is not 0. Again, we suppose that (A(q) = a naught + a1q + a2q squared +...). Okay, let us prove it. So, suppose that. Suppose that B(q) is the inverse of A(q). And B(q) = B naught + b1q +b2q squared +... And then, since we know what that A(q).B(q)=1, we can take the product in the left hand side and say that, all it's coefficients except of the low term r = 0 and the low term is = 1. So this means that and thus a naught b naught = 1. This is the lowest term of this product. The coefficient in front of q1 is a naught b1 + a1b naught. And it should be equal to 0. A naught b2 + a1b1 +a2b naught = 0,.... A naught bn + a1bn- 1 +...+ An b naught, will be also = 0. So we'll have a system move into [INAUDIBLE] equations or that type. I claim that all bs are uniquely determined from these system of equations, if we know the as, provided that a naught is non-zero. Of course, if a naught is none 0 then, so if a naught is 0, this means that this equality can hold for any a naught. This means that such B(q) does not exist, so if a naught is 0, this means that A(q) does not have an inverse. And what happens if it is not 0? This means that. B naught is just 1/ a naught. From the first equation, we find that b naught = 1 / a naught. But in the second equation. We find b1 it is equal to. A1 b naught / a naught. We can divide by a naught because it is not 0. and we already know b naught from the first equation, so b2 =- a1b1 + a2 b naught / a naught and so on. So we can find one by one, all the coefficients in this inverse. And you see that they are uniquely determined from this system of equations and so, we can find the expression for the inverse of A or the power series of A(q). [MUSIC]