Answer
$${\text{The series diverges}}$$
Work Step by Step
$$\eqalign{
& \frac{1}{{12}} + \frac{1}{6} + \frac{1}{3} + \frac{2}{3} + \cdots \cr
& \frac{{1/6}}{{1/12}} = 2,\,\,\,\,\frac{{1/3}}{{1/6}} = 2,\,\,\,\frac{{2/3}}{{1/3}} = 2 \cr
& {\text{This is a geometric series with }}{a_1} = \frac{1}{{12}}{\text{ and }}r = 2 \cr
& {\text{The sum of the series is }}{S_\infty } = \frac{{{a_1}}}{{1 - r}}\,\,\left( {{\text{where }}\left| r \right| < 1} \right) \cr
& \left| r \right| = \left| 2 \right| = 2 > 1 \cr
& {\text{The series diverges}} \cr} $$
