There is a technique we developed for series acceleration using the Wilf-Zeilberger method. Here is a simple Maple code in this regard, you need to download this too.

The idea is you start with a WZ-pair $(F,G)$ and you get an infinite family of hypergeometric series that help compute your constant. Take for example, the Riemann zeta $\zeta(3)=\sum_{n=1}^{\infty}\frac1{n^3}$ which is slow in its rate of convergence. We are able to generate an endless list, call these $s$-step accelerations. The first few include:
$$\zeta(3)=\frac52\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{\binom{2n}n\binom{n}nn^3} \tag {1-step}$$
$$\zeta(3)=\frac14\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{\binom{3n}n\binom{2n}nn^3}\frac{56n^2-32n+5}{(2n-1)^2} \tag {2-step}$$
$$\zeta(3)=\frac1{18}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{\binom{4n}n\binom{3n}nn^3}\frac{5265n^4-7182n^3+3717n^2-828n+68}{(3n-1)^2(3n-2)^2} \tag {3-step}$$
This continues *ad infinitum*.

(1) At each step, we gain large binomial denominator (desirable);

(2) At each step, we incur a rational polynomial with increasing degrees.

Question.In view of the competing gain-loss, (1) and (2), does the computational complexity or efficiency improve or gets worse?

*Caveat.* I don't know much about Complexity Theory, hence my question here.
Thank you for help.