Apéry's Rational Approximation to His Constant
Apéry's Rational Approximation to His Constant
R. Apéry used a rapidly converging rational approximation to to prove its irrationality. Both numerator and denominator satisfy the same recurrence equation -(2k+1)(17+17k+5)+=0, with initial conditions =1, =5, =0, and =6. Approximately three decimal digits are gained with each degree.
ζ(3)
n
k
d
k
2
(k+1)
u
k+1
2
k
u
k
3
k
u
k-1
d
0
d
1
n
0
n
1