Question

The series $\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...to\infty$ has the sum=?

• A. $1$
• B. $\frac{2}{3}$
• C. $\infty$
• D. $2$

Answer 1

Answer: (A)

$1$

$s_n=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...\Rightarrow s_n={\sum }_{r=1}^n\frac{1}{r(r+1)}={\sum }_{r=1}^n(\frac{1}{r}-\frac{1}{r+1})$
Hence
${lim}_{n\to \infty }{s}_n={lim}_{n\to \infty }{\sum }_{r=1}^n(\frac{1}{r}-\frac{1}{r+1})={lim}_{n\to \infty }(1-\frac{1}{n+1})=1$