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

Answer

N/a

Explanation

nth term
`T_(n) = (1)/(("nth term of " 1,2,3,….)("nth term of " 2, 3, 4,…))`
`= (1)/({1 + (n - 1) * 1}{2 + (n-1) * 1})`
`= (1)/(n(n + 1))`
`= ((n + 1) - n)/(n(n+1)) = (1)/(n) - (1)/(n+1)`
Put n = 1, 2, 3,…., n on both sides,
`T_(1) = 1 - (1)/(2)`
`T_(2) = (1)/(2) - (1)/(3)`
`T_(3) = (1)/(3) - (1)/(4)`
`vdots " " vdots " " vdots`
`T_(n) = (1)/(n) - (1)/(n + 1)`
Adding columnwise `S_(n) = 1 - (1)/(n + 1) = (n)/(n + 1)cdot`