  This is an expanded version of our earlier paper. Let the $n$th partial sum
of the Taylor series $e = \sum_{r=0}^{\infty} 1/r!$ be $A_n/n!$, and let
$p_k/q_k$ be the $k$th convergent of the simple continued fraction for $e$.
Using a recent measure of irrationality for $e$, we prove weak versions of our
conjecture that only two of the partial sums are convergents to $e$. A related
result about the denominators $q_k$ and powers of factorials is proved. We also
show a surprising connection between the $A_n$ and the primes 2, 5, 13, 37,
463. In the Appendix, we give a conditional proof of the conjecture, assuming a
second conjecture we make about the zeros of $A_n$ and $q_k$ modulo powers of
2. Tables supporting this Zeros Conjecture are presented and we discuss a
2-adic reformulation of it.
