Integration of the greatest integer function 1

The given integral is equivalent to: $I=\int\limits_\frac{1}{2}^1\left(-1\right)^1dx+\int\limits_\frac{1}{3}^\frac{1}{2}\left(-1\right)^2dx+\int\limits_\frac{1}{4}^\frac{1}{3}\left(-1\right)^3dx+\int\limits_\frac{1}{5}^\frac{1}{4}\left(-1\right)^4dx+...$ This simplifies to: $I=(-1+\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+ (-\frac{1}{3}+\frac{1}{4})+(\frac{1}{4}-\frac{1}{5})+...$ $I=-1+2(\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+...)$ $I=-1+2(1-ln2)=-0.3863...$ Finally, $|I|=0.3863...$