Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 8 / b / ii / Solution

In the Cauchy product, $a_j{x}^j$ combines with $b_{i-j}{x}^{i-j}$ to give a term of degree $i$. HenceThis coefficient identity holds for formal power series. For convergent Taylor series it also gives the function product throughout the common open interval of convergence, where absolute convergence justifies regrouping.