There have been various questions on this site about whether there is any upper limit to how fast a series sum of rational numbers can converge on another rational. With the right choice of particular series there appears to be no upper limit.
The most common example more generally is the infinite geometric series
$$\frac{1}{1-z}=z^0+z^1+z^2+...$$
However a much faster converging series is $$\frac{1}{z}=\sum_{k=1}^\infty \frac{1}{(k+z)\prod_{n=0}^{k-2}(n+z)}\tag{1}$$ which can be proved by the telescoping of the partial fraction
$$\frac{1}{z}=\frac{1}{z\left(1+z \right)}+\frac{1}{\left(1+z \right)}$$
These two series can be combined to give the double series $$\frac{z}{z-1}=\sum_{m=0}^\infty \sum_{k=1}^\infty \frac{1}{(k+z^m)\prod_{n=0}^{k-2}(n+z^m)}$$
Are there known examples of generally applicable infinite series for any given rational, that converge faster than (1)?
