  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 $e,\sin1$, 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, Sierpi\'{n}ski, and
Thue-Siegel-Roth. We also make three conjectures.
  (This manuscript was submitted posthumously. The author passed away on
January 16, 2020.)
