Van Wijngaarden transformation

Remarks

By induction, $s_{j,k} = 2^{-j} \sum_{i=0}^{j} \binom{j}{i} s_{k+i}$, a binomially weighted mean of $j + 1$ consecutive partial sums. Example: for the Leibniz series, $s_0, \dots, s_3$ are $1, 0.6667, 0.8667, 0.7238$. The next rows are $0.8333, 0.7667, 0.7952$, then $0.8000, 0.7810$, then $0.7905$, which is within $0.006$ of $\frac{\pi}{4} \approx 0.7854$. Non-example: the series $\sum 2^{-k}$ is not alternating, and its partial sums $1, 1.5, 1.75$ give the averages $1.25, 1.625$. These are worse than the last partial sum; the method relies on the alternation of signs.