Question

The expression $\frac{1}{x+1}+\frac{1}{2(x+1{)}^2}+\frac{1}{3(x+1{)}^3}+...$ is equal to

• A. $\log \left[\frac{x}{x+1}\right]$
• B. $\log \left[\frac{x+1}{x}\right]$
• C. $\log \left[\frac{x+1}{x-1}\right]$
• D. $\log \left[\frac{x-1}{x+1}\right]$

Answer 1

The correct option is B $\log \left[\frac{x+1}{x}\right]$

$\frac{1}{x+1}+\frac{1}{2(x+1{)}^2}+\frac{1}{3(x+1{)}^3}+...$
$=-\log (1-\frac{1}{x+1})$
$=-\log \left(\frac{x}{x+1}\right)$
​​​​​​​$=\log \left(\frac{x+1}{x}\right)$