Question

$\frac{x-y}{x}+\frac{1}{2}(\frac{x-y}{x}{)}^2+\frac{1}{3}(\frac{x-y}{x}{)}^3+\dots .=$

• A. ${\log }_e(x+y)$
• B. ${\log }_e\frac{x}{y}$
• C. ${\log }_e(x-y)$
• D. ${\log }_e\frac{y}{x}$

Answer 1

Answer: (B)

${\log }_e\frac{x}{y}$

Let $S=\frac{x-y}{x}+\frac{1}{2}(\frac{x-y}{x}{)}^2+\frac{1}{3}(\frac{x-y}{x}{)}^3+\dots .$
We know that , $\log (1-X)=-\sum \frac{X^n}{n}$
In this case, $X=\left(\frac{x-y}{x}\right)$
$S=-\log \left(1-\frac{(x-y)}{x}\right)$
$=-\log \left(\frac{y}{x}\right)$
$S=\log \left(\frac{x}{y}\right)$