Could someone give me a hint on how to prove that the spectral norm of an $n×n$ matrix $A = (a_{ij})$, where $a_{ii} = 0$ and $a_{ij} =1/(i-j)$, is smaller than $\pi$.
$$B=\begin{bmatrix}0 & -1 & -\frac{1}{2} & -\frac{1}{3} & \cdots\\1 & 0 & -1 & -\frac{1}{2} \\ \frac{1}{2} & 1 & 0 & -1 \\ \vdots & &&& \ddots \end{bmatrix}$$