Irrationality and Transcendence of Alternating Series Via Continued Fractions

Abstract

Euler gave recipes for converting alternating series of two types, I and II, into equivalent continued fractions, i.e., ones whose convergents equal the partial sums. A condition we prove for irrationality of a continued fraction then allows easy proofs that $e,\sin 1$, and the primorial constant are irrational. Our main result is that, if a series of type II is equivalent to a simple continued fraction, then the sum is transcendental and its irrationality measure exceeds $2$. We construct all ${\aleph }_0^{{\aleph }_0}=\mathfrak{c}$ such series and recover the transcendence of the Davison--Shallit and Cahen constants. Along the way, we mention $\pi$, the golden ratio, Fermat, Fibonacci, and Liouville numbers, Sylvester's sequence, Pierce expansions, Mahler's method, Engel series, and theorems of Lambert, Sierpiński, and Thue-Siegel-Roth. We also make three conjectures. (This manuscript was submitted posthumously. The author passed away on January 16, 2020.)

17 pages