Question

The value of $1+\frac{1}{3^2}+\frac{1.4}{1.2}\frac{1}{3^4}+\frac{\mathrm{1.4.7}}{\mathrm{1.2.3}}\frac{1}{3^6}+\dots$ is

• A. ${\left(\frac{3}{2}\right)}^{\frac{1}{3}}$
• B. ${\left(\frac{5}{4}\right)}^{\frac{1}{3}}$
• C. ${\left(\frac{3}{2}\right)}^{\frac{1}{6}}$
• D. None of these

Answer 1

Answer: (A)

Let $S=1+\frac{1}{3^2}+\frac{1.4}{1.2}\frac{1}{3^4}+\frac{\mathrm{1.4.7}}{\mathrm{1.2.3}}\frac{1}{3^6}+\dots$
$\Rightarrow S=1+\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)+\frac{\frac{1}{3}\cdot \frac{4}{3}}{1.2}{\left(\frac{1}{3}\right)}^2+\frac{\frac{1}{3}\cdot \frac{4}{3}\cdot \frac{7}{3}}{\mathrm{1.2.3}}{\left(\frac{1}{3}\right)}^3+\dots$
$\Rightarrow S=1+\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)+\frac{\frac{1}{3}\left(1+\frac{1}{3}\right)}{2!}{\left(\frac{1}{3}\right)}^2$
$+\frac{\frac{1}{3}\left(1+\frac{1}{3}\right)\left(2+\frac{1}{3}\right)}{3!}{\left(\frac{1}{3}\right)}^3+\dots$
$={\left(1-\frac{1}{3}\right)}^{-\frac{1}{3}}={\left(\frac{2}{3}\right)}^{-\frac{1}{3}}={\left(\frac{3}{2}\right)}^{\frac{1}{3}}$