We note:

$$E\_n = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + ... + \frac{1}{n!}, n\geq0.$$

It is known that $$\lim\_{n \to \infty} E\_n = e.$$

How to prove that $$\lim\_{n \to \infty} \sum\_{k=0}^n (e- E\_k) = e?$$

(We can express this sum by $ E\_n $?).