# Convergence product theorem

1. Dec 18, 2008

### 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?

2. Dec 18, 2008

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.

3. Dec 18, 2008

4. Dec 19, 2008

### 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)