Sum

Evaluate the sum
$\dfrac{1}{3} + \dfrac{2}{3^2} +\dfrac{1}{3^3} + \dfrac{2}{3^4} +\dfrac{1}{3^5} + \dfrac{2}{3^6} + \dots$

$\begin{array}{|rcll|} \hline s &=& \dfrac{1}{3} + \dfrac{2}{3^2} +\dfrac{1}{3^3} + \dfrac{2}{3^4} +\dfrac{1}{3^5} + \dfrac{2}{3^6} + \dots \\\\ &=& \dfrac{1}{3}*\dfrac{3}{3} + \dfrac{2}{3^2} +\dfrac{1}{3^3}*\dfrac{3}{3} + \dfrac{2}{3^4} +\dfrac{1}{3^5}*\dfrac{3}{3} + \dfrac{2}{3^6} + \dots \\\\ &=& \dfrac{5}{3^2} + \dfrac{5}{3^4} + \dfrac{5}{3^6} + \dots \qquad \text{GP:}~ a=\dfrac{5}{3^2},~ r = \dfrac{1}{3^2} \\\\ s&=& \dfrac{5}{3^2} *\left( \dfrac{1}{1-\dfrac{1}{3^2}}\right) \\\\ s&=& \dfrac{5}{9} * \dfrac{9}{8} \\\\ \mathbf{s}&=& \mathbf{\dfrac{5}{8}} \\ \hline \end{array}$