example of Cauchy multiplication rule

Let us form the Taylor expansionMathworldPlanetmath of  exsiny  starting from the known Taylor expansions


and multiplying these series with Cauchy multiplication rule. As power seriesMathworldPlanetmath, both series are absolutely convergent for all real (and complex) values of x and y. The rule gives immediately the series

y+(-y33!+xy)+(y55!-xy33!+x2y2!)+(-y77!+xy55!-x2y32!3!+x3y3!)+(y99!-xy77!+x2y52!5!-x3y33!3!+x4y4!)+ (1)

The parenthesis expressions here seem a bit irregular, but we can regroup and rearrange the terms in new parentheses:

exsiny=y+xy1!1!+(x2y2!1!-y33!)+(x3y3!1!-xy31!3!)+(x4y4!1!-x2y32!3!+y55!)+ (2)

It’s clear that the last series precisely the same terms as the preceding one. The regrouping and the rearranging of the terms is allowable, since also (1) is converges absolutely. In fact, if one would multiply the series of ex with the series y+y33!+y55!+ of sinhy (which converges absolutely x), one would get the series like (1) but all signs “+”; by the Cauchy multiplication rule this series converges especially for each positive x and y, in which case it is a series with positive terms; hence (1) is absolutely convergent.

The form (2) can be obtained directly from the Taylor series formula (http://planetmath.org/TaylorSeries).

Title example of Cauchy multiplication rule
Canonical name ExampleOfCauchyMultiplicationRule
Date of creation 2013-03-22 17:29:19
Last modified on 2013-03-22 17:29:19
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Example
Classification msc 41A58
Classification msc 40-00
Classification msc 30B10
Classification msc 26A24
Synonym Taylor series gotten by multiplication
Related topic HyperbolicFunctions