Cauchy product

If you multiply two finite sums, you collect all the cross-products and add them up. The Cauchy product is the natural way to do the same for two infinite series: you gather all products a_i b_j and organize them by the diagonals where i + j is constant, exactly as the coefficients of polynomials (or power series) multiply.

Given two series sum a_n and sum b_n, their Cauchy product is the series sum c_n where c_n = a_0 b_n + a_1 b_{n-1} + ... + a_n b_0 = sum_{k=0}^n a_k b_{n-k}. The hope is that if sum a_n = A and sum b_n = B, then sum c_n = AB. This is the rule that makes multiplication of power series term-by-term legitimate, since the Cauchy product of sum a_n x^n and sum b_n x^n produces sum c_n x^n.

The honest hypotheses matter. Mertens' theorem guarantees sum c_n = AB provided both series converge and at least one converges absolutely. If both are only conditionally convergent, the Cauchy product can fail to converge at all — a classic counterexample uses a_n = b_n = (-1)^n / sqrt(n+1). When both factors converge absolutely, the product converges absolutely to AB with no caveats.