Q53 / 160 · TS EAMCET 2007 · Mathematics

$\frac{1}{2}-\frac{1}{2 \cdot 2^2}+\frac{1}{3 \cdot 2^3}-\frac{1}{4 \cdot 2^4}+\ldots$ is equal to

  • A
    $\frac{1}{4}$
  • B
    $\log _3\left(\frac{3}{4}\right)$
  • Correct answer
    C
    $\log _e\left(\frac{3}{2}\right)$
  • D
    $\log _e\left(\frac{2}{3}\right)$

Solution

Given, $\frac{1}{2}-\frac{1}{2 \cdot 2^2}+\frac{1}{3 \cdot 2^3}-\frac{1}{4 \cdot 2^4}+\ldots$ On Comparing with $\log _e(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\ldots \infty$ Put $x=\frac{1}{2}$ on both sides, we get $\begin{aligned} \frac{1}{2}-\frac{1}{2 \cdot 2^2}+\frac{1}{3 \cdot 2^3}-\frac{1}{4 \cdot 2^4}+\ldots & =\log _e\left(1+\frac{1}{2}\right) \\ & =\log _e\left(\frac{3}{2}\right)\end{aligned}$

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

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