fractional part of Riemann zeta $\sum_{s=2}^\infty \{\zeta (s)\}=1$ 

$$\sum_{s=2}^\infty \{\zeta (s)\}=1$$
where $\zeta (s)$ is Riemann zeta, $\{x\}$ denotes the fractional part of the real number $x$


The problem was proposed by Michael Th. Rassias
$\{\zeta(2)\}=\frac{\pi^2}6-1,$ How to go on?
Thanks a lot!

Since $ 1 < \zeta(s) <2$ for $s \ge 2$, it's equivalent to showing that $ \displaystyle\sum_{s=2}^{\infty} \left( \zeta(s)-1 \right)= 1$.
In which case,
$$ \sum_{s=2}^{\infty} \left( \zeta(s)-1 \right)= \sum_{s=2}^{\infty}\sum_{n=2}^{\infty} \frac{1}{n^{s}} = \sum_{n=2}^{\infty} \sum_{s=2}^{\infty} \frac{1}{n^{s}}$$
$$ = \sum_{n=2}^{\infty} \frac{\frac{1}{n^{2}}}{1-\frac{1}{n}} = \sum_{n=2}^{\infty} \frac{1}{n(n-1)}$$
$$=\sum_{n=2}^{\infty} \left( \frac{1}{n-1} - \frac{1}{n}\right) $$
$$ = \lim_{N \to \infty} \left(1- \frac{1}{2} + \frac{1}{2} -\frac{1}{3} + \ldots + \frac{1}{N-1} - \frac{1}{N} \right)$$
$$ = 1- \lim_{N \to \infty}\frac{1}{N} = 1$$
