Is there a closed form for $\sum_{k=0}^n \frac{x^k}{k!}$?

Question

What is the closed form of $$\sum_{k=0}^n \frac{x^k}{k!}$$ as a function of $x$ and $n$?

Knowing that it converges to $e^x$ when $n\to \infty$.

Answer 1

$$\sum_{k=0}^n \frac{x^k}{k!}=\frac{ \Gamma (n+1,x)}{n!}e^x$$ where appears the incomplete gamma function.

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