Originally Posted by

**davidmccormick** Consider the function

$\displaystyle f(s) = \sum_{r=1} ^ {\infty} \frac{1}{r^s}$.

I have so far managed to show that the series converges for each $\displaystyle s\in\ (1,{\infty})$ and that this series defines a continuous function $\displaystyle f : (1,{\infty}) \rightarrow\mathbb{R} $. I am however struggling to show that:

(i) $\displaystyle f$ is differentiable and that $\displaystyle f'(s) < 0$ for all $\displaystyle s \in\ (1,{\infty})$.

(ii) $\displaystyle f$ is differentiable and that $\displaystyle f''(s) > 0$ for all $\displaystyle s \in\ (1,{\infty})$.

Any help would be greatly appreciated.

thanks