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Apéry's Rational Approximation to His Constant

degree
0
n
0
d
0
= 0
ζ(3)-
n
0
d
0
= 1.202056903
R. Apéry used a rapidly converging rational approximation to
ζ(3)
to prove its irrationality. Both numerator
n
k
and denominator
d
k
satisfy the same recurrence equation
2
(k+1)
u
k+1
-(2k+1)(17
2
k
+17k+5)
u
k
+
3
k
u
k-1
=0
, with initial conditions
d
0
=1
,
d
1
=5
,
n
0
=0
, and
n
1
=6
. Approximately three decimal digits are gained with each degree.
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