Solution
= Solution
Subtract the right side from the left and write $g(x)=1-(1+x)^{-p}-px(1+x)^{-p-1}$. Then $g(0)=0$ and $g'(x)=p(p+1)x(1+x)^{-p-2}>0$ for $x>0$, so the strict inequality follows. Set $x=1/n$ and multiply by $n^{-p}>0$ to obtain
$$
\boxed{n^{-p}-(n+1)^{-p}>p(n+1)^{-p-1},\qquad p,n>0.}
$$
This decreasing-power difference supplies a <telescoping bound for a p-series>.