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Find the sum of the series: (1)/((1...

Find the sum of the series:
`(1)/((1xx3))+(1)/((3xx5))+(1)/((5xx7))+...+(1)/((2n-1)(2n+1))`

Text Solution

Verified by Experts

`S_(n) = (1)/(1xx3) + (1)/(3xx5) + (1)/(5xx7) + …`
` T_(n) = (1)/((2n-1)(2n +1))`
` = (1)/(2) ((1)/(2n-1)-(1)/(2n+1))`
` T_(1) = (1)/(2) ((1)/(1)-(1)/(3))`
` T_(2) = (1)/(2) ((1)/(3) - (1)/(5))`
` T_(3) (1)/(2) ((1)/(5) -(1)/(7))`
`T_(n) = (1)/(2) ((1)/(2n-1)-(1)/(2n+1))`
`S_(n) = (1)/(2) (1-(1)/(2n+1)) = (n) /(2n+1)`
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