Inequality $(n+1)^{-s} \leq (2n)^{-s}$ true for all $s\leq1$ and natural $n$?

Question

On the line $S_{2n}-S_n$ I don't understand how the first inequality was established for $s \leq 1$. I see how it works for $0 \leq s \leq 1$ but not s < 0. Any clues?

Answer 1

You cannot use this inequality if $s<0$. Note that for $s=0$ the equality in that passage holds. But if $s<0$ the divergence of the series is quite trivial. For example, rewrite $$\frac{1}{\left(n+1\right)^{s}}+\dots+\frac{1}{\left(2n\right)^{s}}$$ and using the fact $s<0$.