Cauchy Product of Series

Given two geometric power series A(z) = Σi=0∞ (rz)i and B(z) = Σj=0∞ (sz)j, their Cauchy product groups terms along anti-diagonals of constant total degree n = i + j:

Interactive Experiment

Compare diagonal and square partial sums across sign cancellations in the Cauchy Product of Series interactive workspace:

1. Start with the default Ratio r = 1/2, Ratio s = 1/2, Probe z = 1, and Degree N = 2. Verify Coefficient cN = 0.75, Cauchy partial sum = 2.75, and Product of finite sums = 3.0625.
2. Set Ratio s = -1/2 and Degree N = 4 (keeping Ratio r = 1/2, Probe z = 1). Notice that odd anti-diagonals cancel to zero while even anti-diagonals give c4 = 1/16 = 0.0625 and C4 = 21/16 = 1.3125 approaching 4/3.
3. Set Ratio r = 0, Ratio s = 1/2, Probe z = 1/2, and Degree N = 2. Because the first series is the constant 1, the triangular and square regions contain the same nonzero column, so Cauchy partial sum and Product of finite sums both equal 21/16 = 1.3125.
4. Test Degree N = 0 (where C0 = 1 differs from the infinite product 4 by 3) and Probe z = 0 (where only c0 = 1 contributes to the evaluated sum).