Pretty good approximation for Pi

[cuallito]

So \sqrt[5]{306} is a pretty good approximation for Pi (=3.14155).

If you add 1/51, so that you have \sqrt[5]{306+1/51} you get 3.1415925 (last digit is 6 for actual Pi.)

If you add 1/12997, \sqrt[5]{306+1/51+1/12997} you get 3.141592653587 (vs 3.141592653589 for actual Pi.)

And so on. As you can see it converges pretty rapidly!

I was wondering if there was a similar series for Pi^5 that has already been discovered?

 

[CompuChip]

Is there some pattern? I.e. why 306, and can I easily see what comes after 51, 12997, ... ? Because if they are just "random" numbers then this is nice, but it won't be more useful than just learning decimals of pi.