Q63 / 170 · Manipal (MET) 2021 · Mathematics

The value of $\sum_{r=1}^{\infty}\left[3 \cdot 2^{-r}-2 \cdot 3^{1-r}\right]$ is

  • A
    2
  • B
    $1 / 2$
  • C
    1
  • Correct answer
    D
    0

Solution

$\sum_{r=1}^{\infty}\left[3 \cdot 2^{-r}-2 \cdot 3^{1-r}\right]=3 \sum_{r=1}^{\infty} \frac{1}{2^r}-2 \sum_{r=1}^n 3^{1-r}$ $\begin{aligned} & =3\left[\frac{1}{2}+\frac{1}{2^2}+\ldots \infty\right]-2\left[1+\frac{1}{3}+\frac{1}{3^2}+\ldots \infty\right] \\ & =\frac{3}{2}\left[1+\frac{1}{2}+\frac{1}{2^2}+\ldots\right]-2\left[1+\frac{1}{3}+\frac{1}{3^2}+\ldots \infty\right] \\ & =\frac{3}{2}\left(\frac{1}{1-\frac{1}{2}}\right)-2\left(\frac{1}{1-\frac{1}{3}}\right)=\frac{3}{2} \times 2-\frac{2 \times 3}{2}=0\end{aligned}$

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

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