Answer
(a) $\frac{26}{27}$
(b) $\frac{80}{81}$
(c) $\frac{242}{243}$
Work Step by Step
$\displaystyle \sum_{i=1}^{∞}2(\frac{1}{3})^i$
(a) $2(\frac{1}{3})^1+2(\frac{1}{3})^2+2(\frac{1}{3})^3=\frac{2}{3}+\frac{2}{9}+\frac{2}{27}=\frac{18+6+2}{27}=\frac{26}{27}$
(b) $2(\frac{1}{3})^1+2(\frac{1}{3})^2+2(\frac{1}{3})^3+2(\frac{1}{3})^4=\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\frac{2}{81}=\frac{54+18+6+2}{81}=\frac{80}{81}$
(c) $2(\frac{1}{3})^1+2(\frac{1}{3})^2+2(\frac{1}{3})^3+2(\frac{1}{3})^4+2(\frac{1}{3})^5=\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\frac{2}{81}+\frac{2}{243}=\frac{162+54+18+6+2}{243}=\frac{242}{243}$
