<s>[INST] Find the : $\sum_{i=1}^{n}\dfrac{1}{i!}=\text{?}$ 
Find the :
$$\sum_{i=1}^n \frac{1}{i!}=\text{?}$$

For ex :

$$\sum_{i=1}^{100}\dfrac{1}{i!}=\text{?}$$


My Try :
$$\frac{1}{2\times 1}+\frac{1}{1 \times 2 \times 3 }=\frac{3(1)+1}{1 \times 2 \times 3 }$$
$$\frac{4}{1 \times 2 \times 3 }+\frac{1}{1 \times 2 \times 3 \times 4 } = \frac{4(4)+1}{1 \times 2 \times 3 \times 4}$$
$$\frac{17}{1 \times 2 \times 3 \times 4 }+\frac{1}{1 \times 2 \times 3 \times 4\times 5 }=\frac{5(17)+1}{1 \times 2 \times 3 \times 4 \times 5}$$
Now what ?
 [/INST] The "closed form" is $\frac{e \Gamma(n+1,1)}{n!} - 1$ where $\Gamma(\cdot,\cdot)$ is the incomplete Gamma function.
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