Elementary proofs of generalized continued fraction formulae for $e$

Abstract

In this short note we prove two elegant generalized continued fraction formulae $$e=2+\frac{1}{1+\frac{1}{2+\frac{2}{3+\frac{3}{4+\ddots }}}}$$ and $$e=3+\frac{-1}{4+\frac{-2}{5+\frac{-3}{6+\frac{-4}{7+\ddots }}}}$$ using elementary methods. The first formula is well-known, and the second one is newly-discovered in arXiv:1907.00205 [cs.LG]. We then explore the possibility of automatic verification of such formulae using computer algebra systems (CAS's).

4 pages. Comments welcome