Math Help Forum - View Single Post

davidmccormick · #1 November 27th, 2008, 01:06 PM

Consider the function

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

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

Any help would be greatly appreciated.
thanks