Is there a relationship between $e$ and the sum of $n$-simplexes volumes?

When I look at the Taylor series for $e^x$ and the volume formula for oriented simplexes, it makes $e^x$ look like it is, at least almost, the sum of simplexes volumes from $n$ to $\infty$. Does anyone know of a stronger relationship beyond, "they sort of look similar"?

Here are some links:
Volume formula
http://en.wikipedia.org/wiki/Simplex#Geometric_properties

Taylor Series
http://en.wikipedia.org/wiki/E_%28mathematical_constant%29#Complex_numbers

The answer is, it's just a fact “cone over a simplex is a simplex” rewritten in terms of the generating function:

observe that because n-simplex is a cone over (n-1)-simplex $\frac{\partial}{\partial x}vol(\text{n-simplex w. edge x}) = vol(\text{(n-1)-simplex w. edge x})$; in other words $e(x):=\sum_n vol\text{(n-simplex w. edge x)}$ satisfies an equvation $e'(x)=e(x)$. So $e(x)=Ce^x$ -- and C=1 because e(0)=1.