Question

$1+\frac{1}{{3.2}^2}+\frac{1}{{5.2}^4}+\frac{1}{{7.2}^6}+....=$

• A. ${\log }_{e}2$
• B. ${\log }_{e}3$
• C. ${\log }_{e}4$
• D. ${\log }_{e}5$

Answer 1

The correct option is B ${\log }_{e}3$

Given equation $1+\frac{1}{{3.2}^2}+\frac{1}{{5.2}^4}+\frac{1}{{7.2}^6}+....=$

Multiplying and dividing throughout by $2$, we get

$2[\frac{\frac{1}{2}}{1}+\frac{{\left(\frac{1}{2}\right)}^3}{3}+\frac{{\left(\frac{1}{2}\right)}^5}{5}+.....]$

We know that $[\frac{x}{1}+\frac{x^3}{3}+\frac{x^5}{5}+.....]=\log \left[\frac{1+x}{1-x}\right]$

$=\log \left[\frac{1+\frac{1}{2}}{1-\frac{1}{2}}\right]=\log \frac{\frac{3}{2}}{\frac{1}{2}}={\log }_{e}3$