A new proof of the irrationality of ζ(2) and ζ(3) using Padé approximants

Using Padé approximants to the asymptotic expansion of the error term for the series ∑k=1∞ 1k2, ∑k=1∞ 1k3, and ∑k=1∞(− 1)k + 1k2, we recover Apéryʹs sequences which allow one to prove the irrationality of ζ(2) and of ζ(3). The same method applied to the partial sums of ln(1 − t) also proves the irrationality of certain values of the logarithm.