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Published online by Cambridge University Press: 20 November 2018
The question of the irrationality of functions defined by power series, for rational values of the variable, has attracted much attention for over a hundred years. Legendre, in generalizing Lambert's proof of the irrationality of tan x for rational x, proved an important theorem on the irrationality of continued fractions with integer elements. Here we use Legendre's theorem (Lemma 3) to prove that at least one of a certain pair of power series is irrational whenever the variable is rational and satisfies a further condition. We prove the following: