2020/09/30 by Jonathan Sondow, Sondow, Jonathan
Mathematics · #Advanced Mathematical Identities #History and Theory of Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2009.14644
Euler gave recipes for converting alternating series of two types, I and II,\ninto equivalent continued fractions, i.e., ones whose convergents equal the\npartial sums. A condition we prove for irrationality of a continued fraction\nthen allows easy proofs that e,\sin1, and the primorial constant are\nirrational. Our main result is that, if a series of type II is equivalent to a\nsimple continued fraction, then the sum is transcendental and its irrationality\nmeasure exceeds 2. We construct all \ℵ0\ℵ0= mathfrakc such\nseries and recover the transcendence of the Davison--Shallit and Cahen\nconstants. Along the way, we mention \π, the golden ratio, Fermat,\nFibonacci, and Liouville numbers, Sylvester's sequence, Pierce expansions,\nMahler's method, Engel series, and theorems of Lambert, Sierpi 'nski, and\nThue-Siegel-Roth. We also make three conjectures.\n (This manuscript was submitted posthumously. The author passed away on\nJanuary 16, 2020.)\n