Question

The value of $(\frac{1}{3}+\frac{1}{3\cdot 3^3}+\frac{1}{5\cdot 3^5}+\frac{1}{7\cdot 3^7}+.....)$ is

• A. $\frac{1}{2}{\log }_{e}5$
• B. $3{\log }_{e}2$
• C. $\frac{1}{2}{\log }_{e}2$
• D. ${\log }_{e}2$

Answer 1

Answer: (D)

${\log }_{e}2$

The value of $(\frac{1}{3}+\frac{1}{3\cdot 3^3}+\frac{1}{5\cdot 3^5}+\frac{1}{7\cdot 3^7}+.....)$ is
$=x+\frac{x^3}{3}+\frac{x^7}{7}+....\infty$, where $x=\frac{1}{3}$
$=\frac{1}{2}{\log }_e\left(\frac{1+x}{1-x}\right)$
$=\frac{1}{2}{\log }_e\left(\frac{1+\frac{1}{3}}{1-\frac{1}{3}}\right)$

$=\frac{1}{2}{\log }_{e}2$