In our definition, generating functions were formal power series. And now I will recall, or maybe tell you, what are formal power series and how to deal with them. Probably, you have seen power series in your calculus course, or in some other courses, but these were functions. And functions were taking values for some given value of an argument. Formal powers series is something different. It's something like a polynomial, but a polynomial of infinite degree so to say. So a formal definition is as follows, definition, A formal, Power series, Is, Is just an expression, Of the following form. A(q) = a0 + a1q + a2q squared, and so on. Just as in the definition of the generating function, where all ais are, Real numbers, or, This is not very important, you can take them from your favorite ring. Or well you can take, say complex numbers or formal power series from with rational coefficients or whatever you like. So, the set of all formal power series will be denoted by, Formal power series, Is denoted by, R[[q]] in double square brackets. So, you probably know that R[[q]] in usual square brackets stands for polynomials with real coefficients in the variable q. And if we add double brackets here, we allow these polynomials to have, in quotes, infinite degree. So that this, well expression is allowed to have infinitely non-zero coefficients in front of the powers of q. Okay, we do not require that infinitely many of these coefficients are non-zero. So in particular all polynomial's are formal power series. So, R of q is included into the set of formal power series. So, All polynomials are formal power series. Okay, you know how to add and multiply polynomials and how to add and multiply power series. Well the rule is pretty much the same, but well with the only exception that you're dealing with infinitely long expressions. But, say if you want to add two power series you just proceed as if you were proceeding with polynomials. So, how to add and multiply. Formal power series. So addition, if you want to add A(q) + B(q). Yeah I'm using a variable q, but of course you can use any letter you like here. So if you have two formal power series A(q) = a0 + a1q + a2q squared + etc. And B(q), Is b0 + b1q + b2 q squared plus etc. Their sum is defined in absolutely the same way as if these were polynomials. So the sum is a0 + b0 plus q times (a1 + b1)q + (a2 + b2)q squared + and so on, (an + bn)q to the power n + etc. So the sum is just taking coefficient once just as with polynomials. Multiplication is Is also familiar with everyone who, Has seen this on polynomials. So if you want to multiply A of q and B of q, you need to take just these two brackets. A0 plus a1q plus a2q squared plus etc, times b0 plus b1q plus b2q squared plus etc. And you just multiply this brackets and compute the coefficients in front each power of q. Well, the only challenge here is that the number of summons in each bracket is infinite. That's something new. That's something you haven't seen in the polynomial case. So when you were multiplying polynomials, but, okay, let's try to compute say the lowest term. The lowest term is the term where q does not occur at all. So to get the lowest term you need to take a0 from this bracket and b0 from that bracket. So you get, a0 times b0. Okay, what do we get in front of q? To get the first power of q, you need to take either the lowest term here and multiply it by b1q, from that bracket, or you can take a1q from the first and multiply by b0. So you get ( a0b1 + a1b0 ) times q, plus what do we get in front of q squared? We proceed in the similar way and we see that this is equal to a0b2 + a1b1 + a2b0 times q squared. And you see that to find the coefficient in front of this power of q, in front of q squared, we need only finitely many operations. Despite the fact that the number of summons in each bracket is infinite. So the same thing holds for an arbitrary power of q. So to find the coefficient in front of q to the power n we need to take a0 and multiply it by bn sum it up with a1 times bn- 1 + a2 times b- 2 etc. So this is equal to, so we're at etc here plus. A0bn + a1bn- 1 + and so on until we get anb0 times qm + etc. So let me say this again because this is quite an important thing, which we haven't seen in the case of polynomials, to compute each coefficient, we need only finitely many operations to compute each coefficient of the products we need to compute a finite sum of products of coefficients. Despite the fact that we are dealing with infinitely long things. Okay, so now we know how to add and multiply formal power series, and what about division?