Question

The value of $-\left(\frac{4}{5}+\frac{4^2}{5^2.2}+\frac{4^3}{5^3.3}+\frac{4^4}{5^4.4}+\dots \infty \right)$ is

• A. $\log \frac{9}{5}$
• B. $\log \frac{4}{5}$
• C. $\log \frac{5}{4}$
• D. $-\log 5$

Answer 1

The correct option is D $-\log 5$

The expansion for $\log (1-x)$ is
$\log (1-x)=-(x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^5}{5}+....)$

Take $x=\frac{4}{5}$, we get RHS

$=-(\frac{4}{5}+\frac{4^2}{5^2\cdot 2}+\frac{4^3}{5^3\cdot 3}+\frac{4^4}{5^4\cdot 4}+\frac{4^5}{5^5\cdot 5}+....)$

$=\log (1-\frac{4}{5})$

$=\log \left(\frac{1}{5}\right)$

$=-\log 5$ ....... $[\because \log \left(\frac{1}{x}\right)=-\log x]$

Hence, $-(\frac{4}{5}+\frac{4^2}{5^2\cdot 2}+\frac{4^3}{5^3\cdot 3}+\frac{4^4}{5^4\cdot 4}+\frac{4^5}{5^5\cdot 5}+....)=-\log 5$