Question 29
Prove that
$$
\frac{1}{3!}+\frac{5}{4!}+\frac{11}{5!}+\ldots+\frac{n^{2}+n-1}{(n+2)!}\lt \frac{1}{2}
$$
for every natural number $n$. By $n!$ (read "$n$ factorial") we denote the product of the first $n$ nonzero natural numbers: $n!=1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$.
$$
\frac{1}{3!}+\frac{5}{4!}+\frac{11}{5!}+\ldots+\frac{n^{2}+n-1}{(n+2)!}\lt \frac{1}{2}
$$
for every natural number $n$. By $n!$ (read "$n$ factorial") we denote the product of the first $n$ nonzero natural numbers: $n!=1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$.