Question

find the sum to n terms of the series 1/1.2+1/2.3+1/3.4+.......

Answer 1

Given:

$1/1\cdot 2+1/2\cdot 3+1/3\cdot 4+...$

To find:

Sum of n terms of series

Solution:

$a_1=\frac{1}{1\cdot 2}=1-\frac{1}{2}$

$a_2=\frac{1}{2\cdot 3}=\frac{1}{2}-\frac{1}{3}$

$a_3=\frac{1}{3\cdot 4}=\frac{1}{3}-\frac{1}{4}$

:

:

:

:

$a_n=\frac{1}{n}-\frac{1}{n+1}$

Adding all terms of the series then we get

$S_n=a_1+a_2+a_3+...+a_n$

$S_n=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+..+\frac{1}{n}-\frac{1}{n+1}$

$S_n=1-\frac{1}{n+1}$

$S_n=\frac{n+1-1}{n+1}=\frac{n}{n+1}$

Hence, the sum of n terms of given series=$\frac{n}{n+1}$