This is one of those times when FindSequenceFunction
works.
Make a list of the terms. Then let Mathematica find a function, $f$, that generates the coefficients. That is, $f(k) = a_k$, the Taylor series coefficients. We know that $f(k)$ should be $1/k!$, but we want Mathematica to figure that out.
terms = List @@ Normal[Series[Exp[x], {x, 0, 10}]];
f = FindSequenceFunction[terms /. x -> 1]
(* 1/Pochhammer[1, -1 + #1] & *)
Well, that was a surprise. We can get a simpler expression for the coefficients in our summation like this
a[k] = FullSimplify[f[k], k ∈ PositiveIntegers] (* 1/Gamma[k] *)
s[n_] = Inactive[Sum][a[k] x^(k - 1), {k, 1, n}]
$$\underset{k=1}{\overset{n}{\sum }}\frac{x^{k-1}}{\Gamma (k)}$$
We accept the gamma function in lieu of the factorial. We recover the exponential in the limit as $n\rightarrow\infty$
Limit[s[n], n -> ∞] // Activate (* E^x *)