Cauchy Product - Series

Series

A particularly important example is to consider the sequences to be terms of two strictly formal (not necessarily convergent) series

usually, of real or complex numbers. Then the Cauchy product is defined by a discrete convolution as follows.

for n = 0, 1, 2, ...

"Formal" means we are manipulating series in disregard of any questions of convergence. These need not be convergent series. See in particular formal power series.

One hopes, by analogy with finite sums, that in cases in which the two series do actually converge, the sum of the infinite series

is equal to the product

just as would work when each of the two sums being multiplied has only finitely many terms. This is not true in general, but see Mertens' Theorem and Cesàro's theorem below for some special cases.