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The Cauchy product may apply to infinite series [1] [2] or power series. [3] [4] When people apply it to finite sequences [5] or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution).
The Cauchy product of two infinite series is defined even when both of them are divergent. In the case where a n = b n = (−1) n, ...
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions.
When the sequences are the coefficients of two polynomials, then the coefficients of the ordinary product of the two polynomials are the convolution of the original two sequences. This is known as the Cauchy product of the coefficients of the sequences.
The proof is the same as for complex-valued series: use the completeness to derive the Cauchy criterion for convergence—a series is convergent if and only if its tails can be made arbitrarily small in norm—and apply the triangle inequality. In particular, for series with values in any Banach space, absolute convergence implies convergence ...
The Cauchy condensation test is a generalization of this ... and the right equality uses the standard formula for a geometric series. The product is divergent, ...
The set of such Cauchy sequences forms a group (for the componentwise product), and the set of null sequences (sequences such that ,, >,) is a normal subgroup of . The factor group C / C 0 {\\displaystyle C/C_{0}} is called the completion of G {\\displaystyle G} with respect to H . {\\displaystyle H.}
A discrete convolution of the terms in two formal power series turns a product of generating functions into a generating function enumerating a convolved sum of the original sequence terms (see Cauchy product). Consider A(z) and B(z) are ordinary generating functions.