[math-fun] Happy approximate pi day

I see from various friends that it is approximate pi day, 22/7 if you live outside the USA. I’m sure many of us know Dalzell’s classic result that int_0^1 x^4(1-x)^4/(1+x^2) dx = 22/7 - pi which proves that pi is strictly less than 22/7, since the integrand is nonnegative. But for fun, here are a couple of infinite series with positive terms that also equal 22/7 - pi: sum_{n=2}^\infty 60/[(4*n^2-1)(16*n^2-1)(16*n^2-9)] sum_{n=1}^\infty 640n/[(4n+1)(4n+3)(4n+5)(4n+7)(4n+9)(4n+11)] sum_{n=6}^infty 2^{n-3}(n+242)/[15(2n+1)(2n+3)C(2n,n)] Stege