Question

The value of $\frac{1}{2}+\frac{1}{3^2}+\frac{1}{2^3}+\frac{1}{3^4}+\frac{1}{2^5}+\frac{1}{3^6}+\cdots \cdots \infty$ is

• A. $\frac{19}{24}$
• B. $\frac{11}{24}$
• C. $\frac{13}{24}$
• D. $\frac{23}{24}$

Answer 1

The correct option is A $\frac{19}{24}$

We have,
$\frac{1}{2}+\frac{1}{3^2}+\frac{1}{2^3}+\frac{1}{3^4}+\frac{1}{2^5}+\frac{1}{3^6}+\cdots \cdots \infty =(\frac{1}{2}+\frac{1}{2^3}+\frac{1}{2^5}+\cdots \infty )+(\frac{1}{3^2}+\frac{1}{3^4}+\frac{1}{3^6}+\cdots \cdots \infty )=\left(\frac{\left(\frac{1}{2}\right)}{1-\left(\frac{1}{2^2}\right)}\right)+\left(\frac{\left(\frac{1}{3^2}\right)}{1-\left(\frac{1}{3^2}\right)}\right)=\frac{2}{3}+\frac{1}{8}=\frac{19}{24}$