Originally Posted by

**Soroban** Hello, Random Variable!

I can help get you started . . .

$\displaystyle \begin{array}{ccccc}\text{We have:} & S &=& 1 - \dfrac{2}{3} - \dfrac{3}{9} + \dfrac{4}{27} - \dfrac{5}{81} + \hdots \\ \\[-3mm]$$\displaystyle

\text{Multiply by }\frac{1}{3}: & \dfrac{1}{3}S &=& \quad\;\;\; \dfrac{1}{3} - \dfrac{2}{9} + \dfrac{3}{3^3} - \dfrac{4}{3^4} + \hdots\end{array}$

$\displaystyle \text{Subtract: }\qquad\quad\frac{2}{3}S \;\:=\;\;\underbrace{1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} - \frac{1}{81} + \hdots}_{\text{geometric series}}$

The geometric series has its sum: .$\displaystyle \frac{1}{1-\left(\text{-}\frac{1}{3}\right)} \:=\:\frac{1}{\frac{4}{3}} \:=\:\frac{3}{4}$

Hence, we have: .$\displaystyle \frac{2}{3}S \:=\:\frac{3}{4} \quad\Rightarrow\quad S \:=\:\frac{9}{8} \quad\Leftarrow\text{ sum of the series}$