We compute an approximation of using Apéry's recurrence. More precisely, we compute it as . (Remember that tends to .)
To do so, we use the gfun package by Salvy and Zimmermann (Salvy, Bruno and Zimmermann, Paul (1994): Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software , 20 (2):163-177).
The gfun package provides us with a routine for transforming a recurrence equation like
into a procedure. Each of the following procedures encodes the calculation of a sequence given by the equation and its initial values and .
Compute and :
A priori , it is not clear how many digits we can guarantee.
The time used is