Question

$\frac{(a-b)}{a}+\frac{1}{2}(\frac{a-b}{a}{)}^2+\frac{1}{3}(\frac{a-b}{a}{)}^3+.....$ is

• A. $lo{g}_{e}a-lo{g}_{e}b$
• B. $lo{g}_{e}b-lo{g}_{e}a$
• C. $lo{g}_e\left(ab\right)$
• D. $loga+2logb$

Answer 1

The correct option is A $lo{g}_{e}a-lo{g}_{e}b$

given summation is of the form $S={\sum }_{r=1}^{\infty }\frac{x^r}{r}$

$\frac{dS}{dx}=1+x+x^2+x^3...\infty$

$\frac{dS}{dx}=\frac{1}{1-x}$

integrate on both sides,

$S=-\ln \left(1-x\right)$

here $x=(1-\frac{b}{a})$

$S=-\ln (1-(1-\frac{b}{a}\left)\right)=-\ln \frac{b}{a}=\ln \frac{a}{b}=\ln a-\ln b$