Lesson 14

Convergence. Before continuing, we will give an important comment. We usually consider the generating function not as a function but as a formal algebraic object. For example, the formula for the sum of a geometric series \[1+x+x^{2}+x^{3}+\dotsb=\frac{1}{1-x}\] is valid only for \(|x|<1\): in other cases, the series diverges. However, we are not interested in questions of convergence. We will treat the equality above as follows: the objects \(1\) and \(1-x\) are considered as sequences \((1,0,0,\dotsc)\) and \((1,-1,0,0,\dotsc)\). The equality above simply states that the product of two sequences equals a third sequence. The operation of multiplication still needs to be formally defined. We will address this in the next section (it is the usual multiplication of formal power series).