Sketch of Apéry's Proof

Apéry's first remark is that the double sequence

[Maple Math]

tends to [Maple Math] uniformly in [Maple Math] for [Maple Math] in [Maple Math] when [Maple Math] tends to infinity. This stems from the alternating series being uniformly bounded by [Maple Math] . However, the convergence of this series is not strong enough so as to show the irrationality of [Maple Math] . Apéry used summation methods to accelerate the convergence. Namely, define

[Maple Math] and [Maple Math] ,

then [Maple Math] also tends to [Maple Math] . Here appears the crucial recurrence of Apéry: one remarks that it is satisfied by both sequences [Maple Math] and [Maple Math] , with initial conditions

[Maple Math] , [Maple Math] , and [Maple Math] , [Maple Math] .

By a number-theoretic argument, it follows from this recurrence that

[Maple Math] ,


[Maple Math] and [Maple Math] , which is positive.

This is sufficient to prove that [Maple Math] is irrational, and yields an irrationality measure of at least

[Maple Math]