The summation is an incomplete
Maclaurin Series for
exp(n) (see equation 31 on linked page). Since it is incomplete form of
exp(n) is must be always less than
exp(n).
The fact there there is a
(n!)^-1 * n^n term in the sum makes l'Hôpital's Rule difficult to apply.
EDIT: The simplified proof I have above means only that any value of
n less than
infinity will be between 0 and 1. However, the limit can be those. The range of values for the limit:
0≤s≤1 .