Question

1/1*2 +1/2*3 +1/3*4 ....+1/n(n+1) find the sum of the series

Answer 1

Hey There!!!


Here we are asked to find the sum of series: 

$S=\frac{1}{1\times 2}+\frac{1}{2\times 3}+...+\frac{1}{n\times (n+1)}$


Here we see that the general term can be written as:
$T_r=\frac{1}{r\times (r+1)}$ 

Here, there is product of consecutive terms in the denominator. In such cases, we always try to write the General Term as Difference of Two Terms.


Here, we can write:

$T_r = \frac{1}{r}-\frac{1}{r+1}$

Now, the Sum becomes: 

$S=\sum_{r=1}^{n}T_r \\ \\ \\ \implies S=\sum_{r=1}^{n}\frac{1}{r}-\frac{1}{r+1} \\ \\ \implies S=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{n}-\frac{1}{n+1} \\ \\ \\ \implies S=1-\frac{1}{n+1} \\ \\ \\ \implies S=\frac{n+1-1}{n+1} \\ \\ \\ \implies \boxed{S=\frac{n}{n+1}}$


Here in this way, by expressing as a difference, many such terms get cancelled.


Hope it helps,
Purva
Brainly Community