Portal de Periódicos da CAPES

Irrationality and Transcendence of Alternating Series via Continued Fractions

2021; Springer International Publishing; Linguagem: Inglês

10.1007/978-3-030-84304-5_22

2194-1017

Jonathan Sondow,

Functional Equations Stability Results

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.