Question

The value of $\frac{x}{1.2}+\frac{x^2}{2.3}+\frac{x^3}{3.4}+\dots \dots$ is equal to

• A. $\frac{1-x}{x}\log (1-x)$
• B. $\frac{x-1}{x}\log (1-x)$
• C. $\frac{x-1}{x}\log (1+x)$
• D. $1+\left(\frac{1-x}{x}\right)\log (1-x)$

Answer 1

The correct option is D $1+\left(\frac{1-x}{x}\right)\log (1-x)$

Let $y=\frac{x}{1.2}+\frac{x^2}{2.3}+\frac{x^3}{3.4}+\dots$
$\Rightarrow y=\frac{2x-x}{1.2}+\frac{3x^2-2x^2}{2.3}+\frac{4x^3-3x^3}{3.4}+\dots$
$=(x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\dots )-(\frac{x}{2}+\frac{x^2}{3}+\frac{x^3}{4}+\dots )$
$=-\log (1-x)-\frac{1}{x}(\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\dots )$
$=-\log (1-x)-\frac{1}{x}(-\log (1-x)-x)$

$=1+\frac{\log (1-x)}{x}-\log (1-x)$

$=1+\left(\frac{1}{x}-1\right)\log (1-x)$
$=1+\left(\frac{1-x}{x}\right)\log (1-x)$