Question

$\frac{1}{2}-\frac{1}{2^2.2}+\frac{1}{2^3.3}-\frac{1}{2^4.4}+\dots \infty =$

• A. $\log 2$
• B. $\log 3$
• C. $\log \frac{3}{2}$
• D. $\log \frac{2}{3}$

Answer 1

Answer: (C)

$\log \frac{3}{2}$

Let S $=\frac{1}{2}-\frac{1}{2^2.2}+\frac{1}{2^3.3}-\frac{1}{2^4.4}+\dots \infty$
Let $X=\frac{1}{2}$
$\Rightarrow S=X-\frac{X^2}{2}+\frac{X^3}{3}+.....$
We know, $\log (1+x)=\sum (-1{)}^{n+1}\frac{x^n}{n}=x-\frac{x^2}{2}+\frac{x^3}{3}+.....$
Here $x=\frac{1}{2}$
$\Rightarrow S=\log (1+\frac{1}{2})$

$S=\log \left(\frac{3}{2}\right)$