Let $n \in \mathbb{N}_{>0}$ and $s \in \mathbb{R}_{>0}$ then I am interested in the following inequality :
$$\frac{1}{n^s}-\int_n^{n+1} \frac{\mathrm{d}t}{t^s} \leq \frac{s}{n^{s+1}}$$
Here is the way I prove this inequality :
Since $n \geq 1$ then the function : $t \mapsto \frac{1}{t^s}$ is dcreasing on $[n,n+1]$.Hence we have :
$$\frac{1}{n^s}-\int_n^{n+1} \frac{\mathrm{d}t}{t^s} \leq \frac{1}{n^s}-\frac{1}{(n+1)^s}$$
Now using the mean value theorem we get the desired inequality.
I am interested in other proof of this inequality, and if we can find a sharper inequality?