I came across some series many years ago for calculating PI. I found that the first member of that series has been known for a long time in the math world.
It is the set of series defined by:
$$ \pi^k=T\sum_{n=1}^\infty\frac1{n^k}$$
where $[T,k] = [6,2]$, for the well known one.
I have some value pairs for $k$ and $T$ that give rise to other series some of which converge maybe hundreds of times faster than for the above.
For the sake of time, I give the fastest converging member of these series as the one that has $[T,k] = [18243225/2 , 14]$.
For more information about the other series and their $[T,k]$ values, please visit http://gbenro-myinventions.blogspot.com.ng/
How does the case where $[T,k] = [18243225/2 , 14]$ compare with the best algorithms for computing $\pi$ available?
My simple java program on the blog tells me that for just 12 terms, it converges to PI = 3.141592653589793 which is the accuracy available for Math.PI in the Java programming language.
Thanks!
The relative error for $\pi^k$ after summing $n$ terms is $\approx n^{-k}$. Computing the $k$th root then does little change (the relative error becomes $\frac 1kn^{-k}$). Hence the number of correct digits is essentially $k\log_{10}n$. For any fixed $k$, this does not grow very well if we compare it to what the Borweins managed (in the linked Wikipedia article you will find a description of an algorith with nonic convergence, i.e., $\sim 9^n$ correct digits after $n$ steps).