Evaluate the infinite geometric series: $$\frac{3}{2}-\frac{2}{3}+\frac{8}{27}-\frac{32}{243}+\dots$$
This is a geometric series, so I need to find the first term and the common ratio. The first term is $\frac{3}{2}$. To find the common ratio, I can divide any term by the previous term. For example, $\frac{-\frac{2}{3}}{\frac{3}{2}} = -\frac{4}{9}$. I can check that this ratio works for the other terms as well: $\frac{\frac{8}{27}}{-\frac{2}{3}} = -\frac{4}{9}$, $\frac{-\frac{32}{243}}{\frac{8}{27}} = -\frac{4}{9}$. So the common ratio is $-\frac{4}{9}$. To evaluate the infinite series, I need to use the formula $S = \frac{a}{1-r}$, where $a$ is the first term and $r$ is the common ratio. In this case, $a = \frac{3}{2}$ and $r = -\frac{4}{9}$. So the sum is $S = \frac{\frac{3}{2}}{1 - (-\frac{4}{9})} = \frac{\frac{3}{2}}{\frac{13}{9}} = \frac{27}{26}$. 