Closed form of a partial sum of the power series of $\exp(x)$

sequences-and-series combinatorics

I'm not sure you'll like this, but in terms of the incomplete $\Gamma$ function, one can get a closed form as $$\frac{e^{x}\Gamma(n+1,x)}{\Gamma(n+1)}.$$

The incomplete $\Gamma$ function is defined as $$\Gamma(s,x) = \int_x^{\infty} t^{s-1}e^{-t}dt$$.

Related

If a and b are relatively prime and ab is a square, then a and b are squares.
Equation of angle bisector, given the equations of two lines in 2D
Continuous unbounded but integrable functions
What is an example of function $f: \Bbb{N} \to \Bbb{Z}$ that is a bijection?
Prove that $f:\mathbb R^n\to\mathbb R$ is affine if and only if it is convex and concave
The radical of a monomial ideal is also monomial
How to prove that $k[x, xy, xy^2, \dotsc]$ is not noetherian?
How can I find the possible values that $\gcd(a+b,a^2+b^2)$ can take, if $\gcd(a,b)=1$
Solve trigonometric equation: $1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$
What is the solution to the following recurrence relation with square root: $T(n)=T (\sqrt{n}) + 1$?
Generalizing values which Euler's-totient function does not take
How many different permutations are there of the sequence of letters in "MISSISSIPPI"?