$\large\frac{1}{2}-\frac{1}{2, 2^2}+\frac{1}{3.2^3}-\frac{1}{4.2^4}+.........$ is equal to

Question

\[\begin {array} {1 1} (a)\;\frac{1}{4} & \quad (b)\;\log_3 \bigg( \frac{3}{4}\bigg) \\ (c)\;\log_e \bigg(\frac{3}{2}\bigg) & \quad (d)\;\log_e \bigg( \frac{2}{3} \bigg ) \end {array}\]