One of the ways of accelerating the convergence of a series is by transforming into a faster series using optimal parameters. Examples of this approach can be found in this paper. I obtained a generalization of this method to express a series in terms of two more independent variables:

If $|\frac{x}{x+y}| < 1$. Then,

$$ \sum_{n = 1}^{\infty}a_nx^n = \Big(\frac{y}{x+y}\Big)^{r+1} \sum_{n=0}^{\infty}\Big(\frac{x}{x+y}\Big)^{n} \sum_{k=0}^{n} {n+r\choose k+r}a_k y^k $$

This expresses a power series in the LHS in terms of two independent variables $y$ and $r$ which in theory can be fine tuned to make the series converge faster.

Question: How to choose the optimal $y$ and $r$ so that the RHS converges at its fastest rate?

Note: Asked this in MSE but got no replies in a week. Posting it here and deleted it from MSE to avoid duplication.