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 , with initial conditions , , , and . Approximately three decimal digits are gained with each degree. The notation means digits have been left out.
Many other rapidly convergent rational approximations to have been given since Apéry's original article.