[math-fun] Wilf's pi and determinant problems

WDS> Herb Wilf recently died. Here are two open problems he posed. 1. Is there a series for pi=a0+a1+a2+... where a(n) is a rational function of n and a(n-1), which converges faster than geometrically? Equivalently, is there a hypergeometric entire function F(z) with F(1)=pi? --------------- I think you want a0 algebraic. Also, what's special about z=1? But I'm still confused. Suppose a0:=1, a(n):=a(n-1)^2/(n+a(n-1)). What's F? --rwg Do we even have a (presumably AGMish) recurrence a0 = algebraic, a(n) = algebraicfn(n,a(n-1)), a(∞)=π? That MathJax can't be any good. http://www.mathjax.org/demos/mathml-samples/ left the 1/2πi out of Cauchy's integral formula.