Continuing the discussion from the previous post, Ramanujan also gave a continued fraction for as,
where the , starting with n = 1, are given by the linear function,
(Notice the difference from the other version since this one has the cubes twice used as numerators.) Using a similar approach to Apery’s of finding a faster converging version, I found via Mathematica that,
where the are now given by the cubic function,
Of course, a more rigorous mathematical proof is needed that indeed the equality holds.
Update: J.M. from mathstackexchange.com proved that the continued fraction DOES converge to by connecting it to Apery’s version! One consequence of his analysis is that Apery’s generating polynomial can be seen as,