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.

Read more about this topic:  Cauchy Product

Famous quotes containing the word series:

    Depression moods lead, almost invariably, to accidents. But, when they occur, our mood changes again, since the accident shows we can draw the world in our wake, and that we still retain some degree of power even when our spirits are low. A series of accidents creates a positively light-hearted state, out of consideration for this strange power.
    Jean Baudrillard (b. 1929)

    Every man sees in his relatives, and especially in his cousins, a series of grotesque caricatures of himself.
    —H.L. (Henry Lewis)

    The theory of truth is a series of truisms.
    —J.L. (John Langshaw)