Solution
= Solution
In the <Cauchy product>, $a_jx^j$ combines with $b_{i-j}x^{i-j}$ to give a term of degree $i$. Hence
$$
\boxed{c_i=\sum_{j=0}^i a_jb_{i-j}.}
$$
This 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.