Q68 / 170 · Manipal (MET) 2021 · Mathematics

The sum of the series $0.2+0.004+0.00006+0.0000008+\ldots \infty$ is

  • A
    $\frac{200}{891}$
  • Correct answer
    B
    $\frac{2000}{9801}$
  • C
    $\frac{1000}{9801}$
  • D
    None of these

Solution

Sum $=\frac{2}{10}+\frac{4}{10^3}+\frac{6}{10^5}+\frac{8}{10^7}+\ldots \infty$ It is an arithmetic-geometric series. $\begin{aligned} & \therefore \quad S-\frac{S}{10^2}=\frac{2}{10}+\frac{2}{10^3}+\frac{2}{10^5}+\ldots \infty \\ \Rightarrow \quad & \frac{99 S}{100}=\frac{20}{99} \Rightarrow S=\frac{2000}{9801}\end{aligned}$

Hence, the correct answer is option: 'B'.

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