Find the sum of `n` terms`\ ` of the series: `1/(1. 2)+1/(2. 3)+1/(3. 4)+...............+1/(n.(n+1))`

Explanation

`frac{1}{1 cdot 2}+frac{1}{2 cdot 3} cdots+frac{1}{n(n+1)} `


`T_{n}=frac{1}{n(n+1)} T r=frac{1}{r(r+1)} `


`T_{r}=frac{1}{r(r+1)}=frac{1}{r}-frac{1}{r+1}`


`S_{gamma}=sum_{r=1}^{n} T_{r}=sum_{r=1}^{n}(frac{1}{r}-frac{1}{r+1})`


`S_{n}=sum_{r=1}^{n}(frac{1}{r}-frac{1}{r+1}) `


`S_{n}=[frac{1}{1}+frac{1}{2}+frac{1}{3} cdots-frac{1}{n}-frac{1}{2}-frac{1}{3}-frac{1}{4}-frac{1}{n+1}].`


`S_{n}=1-frac{1}{n+1}`


`S_{n}=frac{n+1-1}{n+1}=frac{n}{n+1}`


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