The value of finite sum involving factorials

Question

I have been facing difficulties of solving this finite sum stated below. I know it will be $e^x$ if the summation goes to infinity. how can i deal with if its finite sum. any hint will be highly appreciated

$$\sum_{n=0}^k \frac{x^n}{n!}$$

Answer 1

It equals to $$ e^x-\int_{0}^{x}\frac{t^{k+1}}{\left(k+1\right)!}e^t\text{d}t $$

Answer 2

Using Taylor-Lagrange formula, with the function $x\mapsto e^x$, at the interval $[0,x] $, we find that

$$\sum_{k=0}^n\frac {x^k}{k!}=e^x-\frac {x^{n+1}}{(n+1)!}e^c $$ for some $c\in (0,x) $.