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Convergence product theorem

  1. Dec 18, 2008 #1
    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 [tex]\sum u_n = U[/tex] and a convergent [tex]\sum v_n = V [/tex] . The autor assured that if the difference
    [tex]D_n = \sum_{i=0}^{2n} c_i - U_nV_n,[/tex]
    (where [tex]c_i[/tex] is the Cauchy product of both series and [tex]U_n[/tex] and [tex]V_n[/tex] 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?
     
  2. jcsd
  3. Dec 18, 2008 #2

    statdad

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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.
     
  4. Dec 18, 2008 #3

    statdad

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  5. Dec 19, 2008 #4

    lurflurf

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    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
     
    Last edited: Dec 19, 2008
  6. Dec 22, 2008 #5
    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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