  In this short note we prove two elegant generalized continued fraction
formulae $$e= 2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\ddots}}}}$$
and $$e= 3+\cfrac{-1}{4+\cfrac{-2}{5+\cfrac{-3}{6+\cfrac{-4}{7+\ddots}}}}$$
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).
