Search results
Results from the WOW.Com Content Network
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). Convergence issues are discussed in the next section.
The threefold Cauchy product of 1 − 1 + 1 − 1 + ... is 1 − 3 + 6 − 10 + ..., the alternating series of triangular numbers; its Abel and Euler sum is 1 ⁄ 8. [16] The fourfold Cauchy product of 1 − 1 + 1 − 1 + ... is 1 − 4 + 10 − 20 + ..., the alternating series of tetrahedral numbers , whose Abel sum is 1 ⁄ 16 .
In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. [1] More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other.
6.2 Convolution (Cauchy products) ... A discrete convolution of the terms in two formal power series turns a product of ... we have that f 4 = 4 + 3 · 1 + 2 · 2 ...
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.
The Cauchy product of two series converges to the product of the sums if at least one of the series converges absolutely. ... 1/2 − 1/4 + 1/8 − 1/16 + · · ...
where , is the inner product.Examples of inner products include the real and complex dot product; see the examples in inner product.Every inner product gives rise to a Euclidean norm, called the canonical or induced norm, where the norm of a vector is denoted and defined by ‖ ‖:= , , where , is always a non-negative real number (even if the inner product is complex-valued).
(This limit exists because the real numbers are complete.) This is only a pseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0. But "having distance 0" is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of M.