# 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