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:

    There is in every either-or a certain naivete which may well befit the evaluator, but ill- becomes the thinker, for whom opposites dissolve in series of transitions.
    Robert Musil (1880–1942)

    As Cuvier could correctly describe a whole animal by the contemplation of a single bone, so the observer who has thoroughly understood one link in a series of incidents should be able to accurately state all the other ones, both before and after.
    Sir Arthur Conan Doyle (1859–1930)

    If the technology cannot shoulder the entire burden of strategic change, it nevertheless can set into motion a series of dynamics that present an important challenge to imperative control and the industrial division of labor. The more blurred the distinction between what workers know and what managers know, the more fragile and pointless any traditional relationships of domination and subordination between them will become.
    Shoshana Zuboff (b. 1951)