Here’s a curious series for π that I ran across on Math Overflow.
In case you’re unfamiliar with the notation, n!! is n double factorial, the product of the positive integers up to n with the same parity as n. More on that here.
When n is 0 or −1, n!! is defined to be 1, which is needed above. You could justify this by saying the product is empty, and an empty product is 1. More on that here.
Someone commented on Math Overflow that Mathematica could calculate this sum, so I gave it a try.
f[j_] := (2 j - 3)!! (2 j - 1)!! /((2 j - 2)!! (2 j + 2)!!)
Sum[f[j], {j, 1, Infinity}]
Sure enough, Mathematica returns 2/3π.
It’s a slowly converging series as Mathematica can also show.
In: N[Sum[f[j], {j, 1, 100}]]
Out: 0.210629
In: N[2/(3 Pi)]
Out: 0.212207
If you’re curious about series for calculating π that converge quickly, here’s an algorithm that was once used for a world record calculation of π.