**Holonomic Approach**

We compute an approximation of using Apéry's recurrence. More precisely, we compute it as . (Remember that tends to .)

`> `
**ti[0]:=time():**

`> `
**N:=200:**

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).

`> `
**with(gfun);**

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 .

`> `
**A:=rectoproc({eq,u(0)=0,u(1)=6},u(n));**

`> `
**B:=rectoproc({eq,u(0)=1,u(1)=5},u(n));**

Compute and :

`> `
**a:=A(N);**

`> `
**b:=B(N);**

*A priori*
, it is not clear how many digits we can guarantee.

`> `
**P:=round(evalf(ln(a)/ln(10))*1.1);**

`> `
**Z3[holonomy]:=evalf(a/b,P);**

The time used is

`> `
**ti[holonomy]:=time()-ti[0];**