Question

The sum to infinity of the following series $\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\dots .$. shall be-

• A. $\infty$
• B. 1
• C. 0
• D. None of these

Answer 1

Answer: (B)

Series is $\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\dots \dots \infty$
Here $T_n=\frac{1}{n(n+1)}=\left[\frac{1}{n}-\frac{1}{n+1}\right]$
$\therefore T_1=\frac{1}{1}-\frac{1}{2}$
$\Rightarrow T_2=\frac{1}{2}-\frac{1}{3}$
$T_n=\frac{1}{n}-\frac{1}{n+1}$
$\therefore S_n=T_1+T_2\dots \dots ..+T_n=1-\frac{1}{n+1}$
when $n\to \infty \frac{1}{n+1}\to 0$
$\therefore S_{\infty }=1-0=1$