Basel Problem

The next inequality was derived by Prof.Oshima.

 $A_m=\pi \theta_m=\displaystyle \sum_{k=1}^m\frac{1}{k^2}$,  $B_m=A_m+\dfrac{1}{m+\tfrac{1}{2}}$, $C_m=A_m + \dfrac{1}{m+\tfrac{1}{2}}-\dfrac{1}{12(m^2-\tfrac{1}{4})(m+\tfrac{3}{2})}$

 $\Longrightarrow C_m < A_{\infty} < B_m$