Question

Find the sum of $1+\frac{2}{3}+\frac{4}{9}+\frac{6}{27}+\frac{8}{81}+\frac{10}{243}+\cdots \infty$:

• A. $\frac{2}{3}$
• B. $\frac{5}{2}$
• C. $\frac{19}{45}$
• D. $\frac{81}{17}$

Answer 1

The correct option is B $\frac{5}{2}$

Let $S=1+\frac{2}{3}+\frac{4}{9}+\frac{6}{27}+\frac{8}{81}+\frac{10}{243}+\cdots \infty$
$\therefore 3\left(S\right)=3+2+\frac{4}{3}+\frac{6}{9}+\frac{8}{27}+\frac{10}{81}+\cdots \infty$
$\therefore 2\left(S\right)=4+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\frac{2}{81}+\cdots \infty$
$\therefore 2\left(S\right)=4+2(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\cdots \infty )$
$\Rightarrow 2\left(S\right)=4+2\left[\frac{\frac{1}{3}}{1-\left(\frac{1}{3}\right)}\right]$
$\Rightarrow 2\left(S\right)=4+1$
$\Rightarrow 2\left(S\right)=5$
$\therefore S=\frac{5}{2}$