Understanding the Convergence Product Theorem in Series Algebra

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The discussion centers on the product convergence theorem in series algebra, specifically regarding the Cauchy product of two series. It highlights that if the difference between the Cauchy product and the product of the sums tends to zero, then the product series converges. Participants suggest references for further understanding, including Knopp's treatise and Bromwich's "An Introduction to the Theory of Infinite Series." The conversation emphasizes the relationship between convergence and the Cauchy criterion. Ultimately, the theorem provides a new perspective on analyzing the "tails" of series.
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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?
 
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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.
 
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
 
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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.
 
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