Prove that if $s >1$, then $\displaystyle {\sum_{r=1}^n} \left( {\frac{s}{r}- \frac{1}{r^3}} \right) - s \log n$ tends to a limit as $n \to \infty$ ($s$ being fixed), and that if this limit is $\phi (s)$ then [also show that] $0 \le \left[ {\phi (s) + \frac {1}{s-1}} \right] \le (s-1)$
From A Course of Mathematical Analysis by Shanti Narayan pp.332.
Edit: Solved! $\displaystyle {\lim_{n \to \infty} \sum_{r=1}^n} \left( {\frac{s}{r}- \frac{1}{r^3}} \right) - s \log n$ $= \displaystyle {\lim_{n \to \infty} \sum_{r=1}^n} \left( {\frac{s}{r}} \right) - s \log n -\displaystyle {\lim_{n \to \infty} \sum_{r=1}^n} \left( {\frac{1}{r^3}} \right)$ $=s \gamma - \zeta(3)$ [$\gamma$ := Euler-Mascheroni constant]. Now the inequality, which I suppose to be wrong (but actually it is not), was proved using Mathematical Induction. Thanks to all.