Iterated series

From: Benoit Cloitre (abcloitre_at_wanadoo.fr)
Date: 02/16/05 

Date: 16 Feb 2005 10:45:01 -0500

Defining s_0(n)=\sum_{j=1}{n}a_j where (a_k)_{k>0} are terms of a
convergent series, I define inductively for some suitable sequence
(b_k)_{k>0} the following partial sums involving the remainders of the
previous series :

s_{r+1}(n)=\sum_{j=1}{n}b_j(s_r-s_{r}(j))

where s_{r}=\lim_{n\rightarrow\infty}s_{r}(n)

One of my aims is to determine s_r in closed form for all r for some
sequences (a) and (b). For instance if :

a_k=\frac{1}{k2^k} and b_k=\frac{1}{k}

I obtained :

s_r=\frac{\log(2)^r}{r!}

I'm looking for references of existing work in this small area or
related stuff.

Thanks in advance,
Benoit.