Problem 954. Pi Estimate 2

It seems that the formula is an infinite sum of factorial(n)^2/factorial(2*n+1) (starting at zero) multiplied by
9/(2*sqrt(3)), When (current sum - previous sum) < 10^-n then we should stop the infinite sum. One expected output is the number of summands and the other is our estimated value for pi (rounded to 10 decimal places) in this order. Good luck for anyone trying.

Thanks Rafael!

For the record the stopping condition should use 10^-d, not 10^-n.

More precisely, we aim to utilize the relation
$$ \sum_{n=1}^{\infty} \frac{(n!)^2}{(2n+1)!} = \frac{2\sqrt{3}}{9}\pi-1 $$
to estimate the value of $\pi$ by iteratively adding terms from this series and comparing successive partial sums to a desired level of accuracy.