One of the ways of [accelerating the convergence of a series][1] is by transforming into a faster series using optimal parameters. Examples of this approach can be [found in this paper][2].  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, 
> 
> $$ \sum_{n = 0}^{\infty}a_nx^n = \Big(\frac{y}{x+y}\Big)^{r+1}
 \sum_{n=0}^{\infty}\Big(\frac{x}{x+y}\Big)^{n} \sum_{k=0}^{n}
 {n+r\choose k+r}a_k y^k $$

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?


  [1]: https://en.wikipedia.org/wiki/Series_acceleration
  [2]: https://carma.newcastle.edu.au/jon/Preprints/Papers/Published-InPress/TenProblems/accelerate.pdf

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