Understanding the Convergence Product Theorem in Series Algebra

[thesaruman]

While reading the section about algebra of series of Arfken's essential Mathematical Methods for Physicist, I faced an intriguing demonstration of the product convergence theorem concerning an absolutely convergent series $$\sum u_n = U$$ and a convergent $$\sum v_n = V$$ . The autor assured that if the difference
$$D_n = \sum_{i=0}^{2n} c_i - U_nV_n,$$
(where $$c_i$$ is the Cauchy product of both series and $$U_n$$ and $$V_n$$ are partial sums) tends to zero as n goes to infinity, the product series converges.
The first thing that came in my mind was the Cauchy criterion for convergence, but then I remembered that i and j should be any integer. So I searched through all the net and my books, and didn't find any close idea.
Where should I search for this? Knopp?

 

[statdad]

Knopp's book would be one of my first two choices (if you are referring to his big treatise on series). The other is a similar book by Bromwich. I don't have my copy at home so can't give the exact name, but it too is very good.

 

[statdad]

Here is a link to some other suggestions.

http://www.ericweisstein.com/encyclopedias/books/Series.html

 

[lurflurf]

By Abel's theorem if the sum converges it must converge to UV. So this theorem just combines that with the theorem that if a sequence is Cauchy it must converge. I don't know what the i and j you speak of are. The Bromwich book mentioned by statdad is
An Introduction To The Theory Of Infinite Series (1908)
now back in print and downloadable at books.google.com
http://books.google.com/books?id=ZY...+of+Infinite+Series&ie=ISO-8859-1&output=html

see in Bromwich
Article 33 pages 81-84

 

[thesaruman]

Thank you very much, statdad and lurflurf. I think that I finally understood what the author wanted to say. The product of series presents us to a dramatically new form of seeing the "tails" of the series. By the way, interesting find this google's books. Sorry for my late reply.