In Bailey, Borwein, and Borwein's paper [8] an overview of a method is given for constructing series and algorithms for finding rapid approximations for . Although Ramanujan did not know these algorithms many of the key ingredients are in his notebooks [13]. The algorithms depend crucially on the solvable forms of certain modular equations for the theta-functions due to Ramanujan. In [8] two algorithms are given -- one quartic and one quintic algorithm. In a related paper [5] a septic algorithm is sketched. In [4] a general method is given for constructing p-th order algorithms. These algorithms involve defining a sequence recursively and for which converges to to high order. In general, for us, p-th order convergence of a sequence to means that tends to and that


for some constant C>0. The proof of p-th order convergence depends crucially on identifying as the value of a certain function , which can be defined in terms of elliptic integrals or equivalently in terms of theta-functions.
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