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R. Apéry used a rapidly converging rational approximation to ζ(3) to prove its irrationality. Both numerator

and denominator

satisfy the same recurrence equation

, with initial conditions

, and

. Approximately three decimal digits are gained with each degree.

Many other rapidly convergent rational approximations to ζ(3) have been given since the original article of Apéry.

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Apéry's Rational Approximation to His Constant