Irrationality and Transcendence of Alternating Series Via Continued Fractions

TL;DR

This paper explores the irrationality and transcendence of certain series via continued fractions, providing new proofs and constructing a vast class of such series with implications for famous constants.

Contribution

It introduces a condition for irrationality of continued fractions, proves the transcendence of series equivalent to simple continued fractions, and constructs a large family of such series.

Findings

Proved irrationality of e, sin(1), and the primorial constant.

Established that certain series are transcendental with irrationality measure > 2.

Constructed a continuum of series with these properties.

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 $e,\sin1$, 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, Sierpi\'{n}ski, and Thue-Siegel-Roth. We also make three conjectures. (This manuscript was submitted posthumously. The author passed away on January 16, 2020.)