\displaystyle \mathbf{I} = \sum_{k=1}^{\infty} \left(\frac{1}{2(k+1)}-\frac{1}{2k}\right) + \sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{k+1} \right)- \frac{1}{2} \sum_{k=1}^{\infty} \frac{1}{(k+1)^{2}} = 1 - \frac{\pi^{2}}{12} 