## Task $5 / 79$

Calculate the sum!

$$
s_{n}=1-\frac{1}{2!}-\frac{2}{3!}-\frac{3}{4!}-\ldots-\frac{n-1}{n!}
$$
If we calculate $S_{1}, S_{2}, \ldots, S_{5}$ numerically, we obtain

$$
S_{1}=1 ; \quad S_{2}=\frac{1}{2} ; \quad S_{3}=\frac{1}{6} ; \quad S_{4}=\frac{1}{24} ; \quad S_{5}=\frac{1}{120}
$$

This leads to the conjecture that $S_{n}=\frac{1}{n!}$. If this is true, then

$$
S_{n+1}=S_{n}-\frac{n}{(n+1)!}=\frac{1}{n!}-\frac{n}{(n+1)!}=\frac{n+1-n}{(n+1)!}=\frac{1}{(n+1)!}
$$

Thus, the conjecture is also proven by induction; because the claim holds for $n=1 ; 2 ; \ldots ; 5$, and from the validity for any $n$ follows the validity for $n+1$.