$\lim_{n\to\infty}\left\{n\sum_{k=1}^n\frac{1}{k^5}\right\}$

I need to calculate the limit (if it exist) $$\lim_{n\to\infty}\left\{n\sum_{k=1}^n\frac{1}{k^5}\right\}$$ where $\{x\}$ denotes the fractional part of $x$ and $n\in\mathbb{N}$.

By definition of fractional part function,$$0\leq\left\{n\sum_{k=1}^n\frac{1}{k^5}\right\}<1$$So we have $$0\leq\lim_{n\to\infty}\left\{n\sum_{k=1}^n\frac{1}{k^5}\right\}\leq1$$Sorry but I don't have any ideas. Any help would be highly appreciated.