 Solution

ID: 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

a_{j} x^{j}

combines with

b_{i - j} x^{i - j}

to give a term of degree

i

. Hence

c_{i} = j = 0 \sum i a_{j} b_{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.

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