  Certain numerical methods for initial value problems have as stability
function the nth partial sum of the exponential function. We study the
stability region, i.e., the set in the complex plane over which the nth partial
sum has at most unit modulus. It is known that the asymptotic shape of the part
of the stability region in the left half-plane is a semi-disk. We quantify this
by providing disks that enclose or are enclosed by the stability region or its
left half-plane part. The radius of the smallest disk centered at the origin
that contains the stability region (or its portion in the left half-plane) is
determined for $1\le n\le 20$. Bounds on such radii are proved for $n\ge 2$;
these bounds are shown to be optimal in the limit $n\to +\infty$. We prove that
the stability region and its complement, restricted to the imaginary axis,
consist of alternating intervals of length tending to $\pi$, as $n\to\infty$.
Finally, we prove that a semi-disk in the left half-plane with vertical
boundary being the imaginary axis and centered at the origin is included in the
stability region if and only if $n\equiv 0 \mod 4$ or $n\equiv 3\mod 4$. The
maximal radii of such semi-disks are exactly determined for $1\le n\le 20$.
