**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

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### Famous quotes containing the word series:

“The theory of truth is a *series* of truisms.”

—J.L. (John Langshaw)

“The woman’s world ... is shown as a *series* of limited spaces, with the woman struggling to get free of them. The struggle is what the film is about; what is struggled against is the limited space itself. Consequently, to make its point, the film has to deny itself and suggest it was the struggle that was wrong, not the space.”

—Jeanine Basinger (b. 1936)

“I look on trade and every mechanical craft as education also. But let me discriminate what is precious herein. There is in each of these works an act of invention, an intellectual step, or short *series* of steps taken; that act or step is the spiritual act; all the rest is mere repetition of the same a thousand times.”

—Ralph Waldo Emerson (1803–1882)