Lesson 3

Differentiation. \(\mathcal{A}'(x)\) is the generating function for the sequence \((a_{1}, 2a_{2}, 3a_{3}, \dotsc)\). (We note again that this is just a formal operation, not differentiation in the usual sense: the corresponding series may diverge, and the function may not even be defined. However, if \(\mathcal{A}\) is the Taylor series of some function, the derivative will correspond exactly to this.)

Above, we showed that \(\frac{1}{1-x}\) is the generating function for the sequence \((1,1,1,\dotsc)\), and \(\frac{1}{(1-x)^2}\) is the generating function for the sequence \((1,2,3,\dotsc)\). It is easy to verify that the differentiation operation is consistent with this knowledge.