how I could show that: $\sum_{k=0}^n\binom{n}{k}(-1)^k\frac{n^2+n+1}{nk+n+1}\le1 $?

Question

Show that for all positive integers $ n $ :

$\sum_{k=0}^n\binom{n}{k}(-1)^k\frac{n^2+n+1}{nk+n+1}\le1 $

I would be interest for any replies or any comments

Answer 1

Hint. Check that

$$\sum_{k=0}^{n} \binom{n}{k} (-1)^{k} \frac{n^{2}+n+1}{nk+n+1} = \prod_{k=1}^{n} \left( 1+\frac{1}{nk} \right)^{-1}. $$

The starting idea is to write

$$\sum_{k=0}^{n} \binom{n}{k} (-1)^{k} \frac{n^{2}+n+1}{nk+n+1} = (n^{2}+n+1) \sum_{k=0}^{n} \binom{n}{k} (-1)^{k} \int_{0}^{1} x^{nk+n} \, dx. $$