Among specific topics are Archimedes and the geometric series to be, the binomial series in the hands of Euler, the

Cauchy product, welcome to irrationals: the complete space of real numbers, and features of Ces[sz]ro and Abel means.

From now on, we have to consider odd terms and even terms by using Cauchy product. So, we get to generating terms by dividing the odd terms and the even terms, respectively,

By using Cauchy product in the above equation, we have

We use * to denote the Cauchy product of two species.

The Cauchy product turns Sp into a symmetric monoidal category.

The cycle index series is multiplicative under Cauchy product: if h = k x q, then [Z.sub.h]([x.sub.1], [x.sub.2], ...) = [Z.sub.k]([x.sub.1], [x.sub.2], ...) [Z.sub.q]([x.sub.1], [x.sub.2], ...); see [3, [section] 1.3].

In Section 3, we give necessary and sufficient conditions on a double sequence ([a.sub.k,l]) in order that the Cauchy product double series [[SIGMA].sub.k, l] [a.sub.k,l] * [b.sub.k, l] would be convergent/boundedly convergent/regularly convergent whenever a double series [[SIGMA].sub.k, l] [b.sub.k, l] is convergent/boundedly convergent/regularly convergent.

The Cauchy product of sequences ([a.sub.k]) and ([b.sub.k]) with k [member of] [N.sub.0] is defined to be the sequence ([a.sub.k] * [b.sub.k]), where [a.sub.k] * [b.sub.k]: = [[SIGMA].sup.k.sub.i = 0] [a.sub.i][b.sub.k-i] for k [member of] [N.sub.0], and the Cauchy product of single series [[SIGMA].sup.[infinity].sub.k = 0] [a.s and [[SIGMA].sup.[infinity].sub.k = 0] [b.sub.k] is defined to be the double series [[SIGMA].sup.[infinity].sub.k = 0] [a.sub.k] * [b.sub.k].

and the Cauchy product of double series [[SIGMA].sup.[infinity].sub.k, l = 0] [a.sub.k,l] and [[SIGMA].sup.[infinity].sub.k, l = 0] [b.sub.k, l] is defined to be the double series [[SIGMA].sup.[infinity].sub.k, l = 0] [a.sub.k,l] * [b.sub.k, l].

Then the Cauchy product double series [[SIGMA].sup.[infinity].sub.k,l = 0] [a.sub.k,l] * [b.sub.k, l] is boundedly convergent for every boundedly convergent double series [[SIGMA].sup.[infinity].sub.k, l = 0] [b.sub.k, l] if and only if the double series [[SIGMA].sup.[infinity].sub.k, l = 0] [a.sub.k,l] is absolutely convergent.