Irrationality and Transcendence of Alternating Series via Continued Fractions

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 irrationality a fraction then allows easy proofs that \(e,\sin 1\), primorial constant are irrational. Our main result is that, if type II to simple fraction, sum transcendental its measure exceeds 2. We construct all \(\aleph _0^{\aleph _0}=\mathfrak {c}\) such recover transcendence Davison–Shallit Cahen constants. Along way, mention \(\pi \), golden ratio, Fermat, Fibonacci, Liouville numbers, Sylvester’s sequence, Pierce expansions, Mahler’s method, Engel series, theorems Lambert, Sierpinski, Thue-Siegel-Roth. also make three conjectures.