$\frac{5}{{1.2.3}} + \frac{7}{{3.4.5}} + \frac{9}{{5.6.7}} + ........to\;{\rm{ }}\infty {\rm{ = }}\,{\rm{ - 1 + 3log 2}}$

Question:
$\frac{5}{{1.2.3}} + \frac{7}{{3.4.5}} + \frac{9}{{5.6.7}} + ........to\;{\rm{ }}\infty {\rm{ = }}\,{\rm{ - 1 + 3log 2}}$

$\begin{array}{l}{\rm{Solution: }}\\ - 1{\rm{ + 3log 2}}\\{\rm{ =  - 1 + 3}}\left[ {1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7} + ........to\;{\rm{ }}\infty } \right]\\ = {\rm{ - 1 + }}3 - \frac{3}{2} + \frac{3}{3} - \frac{3}{4} + \frac{3}{5} - \frac{3}{6} + \frac{3}{7} + ........to\;{\rm{ }}\infty \\ = 2 - \frac{3}{2} + \frac{3}{3} - \frac{3}{4} + \frac{3}{5} - \frac{3}{6} + \frac{3}{7} + ........to\;{\rm{ }}\infty \\ = 2 - \frac{3}{2} + \frac{1}{3} + \frac{2}{3} - \frac{3}{4} + \frac{1}{5} + \frac{2}{5} - \frac{3}{6} + \frac{1}{7} + \frac{2}{7} + .......to\;{\rm{ }}\infty \\ = (2 - \frac{3}{2} + \frac{1}{3}) + (\frac{2}{3} - \frac{3}{4} + \frac{1}{5}) + (\frac{2}{5} - \frac{3}{6} + \frac{1}{7}) + .......to\;{\rm{ }}\infty \\ = \frac{5}{{1.2.3}} + \frac{7}{{3.4.5}} + \frac{9}{{5.6.7}} + ........to\;{\rm{ }}\infty \end{array}$