Question

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

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

Answer 1

Answer: (B)

$1$

$\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...\infty$

$\Rightarrow$ The General term of the Given series can be

written as :-

$r^{th}$ term $\Rightarrow T\left(r\right)=\frac{1}{r(r+1)}=\frac{(r+1)-r}{r(r+1)}=\frac{1}{r}-\frac{1}{r+1}$

For $r=1;t_1=\frac{1}{1}-\frac{1}{2}$

$r=2;t_2=\frac{1}{2}-\frac{1}{3}$

$r=n;t_n=\frac{1}{n}-\frac{1}{n+1}$

$S_n=\sum _{r=1}^n{t}_n=\frac{1}{1}-\frac{1}{n+1}=\frac{n}{n+1}$

$\underset{n\to \infty }{lim}\frac{n}{n+1}\Rightarrow \underset{n\to \infty }{lim}\frac{1}{1+\frac{1}{n}}$

$\Rightarrow \underset{n\to \infty }{lim}\frac{1}{1+0}=\frac{1}{1}=1$ (option - B)