I recently came across the sum $$\sum_{n=0}^{\infty} (e- S_n) = e,$$ where $S_n = 1+1+1/2+1/6+...+1/n!$ is the classic $n$th partial sum of $e.$ This gave me an idea. Let $a_{0,0}, a_{0,1}, a_{0,2},\dots$ be a convergent sequence, let $$S_{m,n}=\sum_{k=n}^{\infty} a_{m,k}$$
$$a_{i+1,j} = S_{i,j+1}$$
Now let $b_n=S_{n,0}.$ For the sequence $a_{0,n}=1/n!,$ we have $b_0=e,$ and as I mentioned earlier, $b_1=e.$ I think that $1<b_2<2,$ as the sum decays quickly after the initial term. Also interesting is the fact that if you take $a_{0,n}=0.5^n,$ then $b_{n}=2 \, \forall n,$ as $a_{i,j}=a_{0,j}=2^{-j} \, \forall i,j.$ Perhaps this is the only such sequence with the property that $\{b_n\}$ is constant.
Is there any merit to such sequences? I would like to know if it is possible to determine the long term behavior of $b_n$ for various sequences $\{a_{0,n}\}.$ I believe that someone has already thought of this before, and done extensive research on it, as such an idea seems too simple to have been overlooked.