Does this series acceleration method have a name?

Question

So I recently discovered a fairly simple series acceleration method. It works best for $S=\sum_{m=0}^\infty a_m$ where $a_m=(-1)^mb_m$ and $b_m$ is monotonically decreasing and positive. To start out, we define

$$S_k(n)=\begin{cases}\displaystyle\sum_{m=0}^na_m,&k=0\\\displaystyle\frac{S_{k-1}(n)+S_{k-1}(n+1)}2,&k>0\end{cases}$$

When the following limits exist, the will equal $S$.

$$S=\lim_{n\to\infty}S_k(n)=\lim_{n\to\infty}S_n(k)$$

and this holds for any natural $k$. It is trivial to see how the first limit follows, and it turns out the second limit is an ordinary Euler sum. One can obtain the more general Euler sum be changing the $k>0$ case as follows:

$$S_{k+1}(n)=\frac{S_k(n)+pS_k(n+1)}{1+p}$$

For this reason, I've been calling this a "partial Euler sum", which has the nice advantage of being easy to implement with arguably less floating point errors than directly implementing the Euler sum as well as being much more applicable since you can choose your choices of $n,k$.

I've asked about this in the main mathematics chat room, but it doesn't seem anyone knows the answer to my question:

Does this series acceleration method have a name?

(Please provide a reference)

Answer 1

What you have is the special case of Wynn's formulation of the Euler-Knopp transform:

$$\begin{align*}S_n^{(j)}&=\frac{S_{n-1}^{(j+1)}+q S_{n-1}^{(j)}}{1+q},\;j,n=0,1,\dots\\S_{-1}^{(j)}&=S_{j-1},\;j=0,1,\dots\;(S_{-1}=0)\end{align*}$$

with the specialization $q=1$

---

Here is a Mathematica implementation I posted a while back:

eulerKnopp[seq_?VectorQ, q_: 1] := 
     Module[{n = Length[seq], res, ek},
            ek[1] = 0; res = {First[seq]}; 
            Do[ 
               ek[k + 1] = seq[[k]]; 
               Do[ 
                  ek[j] += (ek[j + 1] - ek[j])/(1 + q);, 
                  {j, k, 1, -1}]; 
               res = {res, ek[1]};, 
               {k, n}]; 
            Flatten[res]]

Use it on the Mercator series:

eulerKnopp[N[Table[Sum[(-1)^j/(j + 1), {j, 0, n}], {n, 0, 10}], 20]]