If $\left|a\right|<1$ and $\left|b\right|<1$, then the sum of the series $a\left(a+b\right)+a^2\left(a^2+b^2\right)+a^3\left(a^3+b^3\right)+\dots$ is

• A

  $\frac{a}{1-a}+\frac{ab}{1-ab}$
• Correct answer
  B

  $\frac{a^2}{1-a^2}+\frac{ab}{1-ab}$
• C

  $\frac{b}{1-b}+\frac{a}{1-a}$
• D

  $\frac{b^2}{1-b^2}+\frac{ab}{1-ab}$

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