Sketch of Apéry's Proof
Apéry's first remark is that the double sequence
tends to uniformly in for in when tends to infinity. This stems from the alternating series being uniformly bounded by . However, the convergence of this series is not strong enough so as to show the irrationality of . Apéry used summation methods to accelerate the convergence. Namely, define
then also tends to . Here appears the crucial recurrence of Apéry: one remarks that it is satisfied by both sequences and , with initial conditions
, , and , .
By a number-theoretic argument, it follows from this recurrence that
and , which is positive.
This is sufficient to prove that is irrational, and yields an irrationality measure of at least