The sum of the series $\frac{1}{1\cdot 2}-\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}-\dots$ upto $\infty$ is equal to:

1. A
   $2{\log }_{e}2$
2. B
   ${\log }_{2}2-1$
3. C
   ${\log }_{e}2$
4. D
   ${\log }_e\left(\frac{4}{e}\right)$