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)
“The professional celebrity, male and female, is the crowning result of the star system of a society that makes a fetish of competition. In America, this system is carried to the point where a man who can knock a small white ball into a series of holes in the ground with more efficiency than anyone else thereby gains social access to the President of the United States.”
—C. Wright Mills (1916–1962)
“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)