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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

`T_(r)=1/((2r-1)(2r+1))`
`1/2(1)/(2r-1)-1/(2r+1)`
`=1/2(V(r-1)-V(r ))," where " V( r)=1/(2r+1)`
`thereforesum_(r=1)^(n)T_(r)=sum_(r=1)^(n)1/2(V(r-1)-V(r ))`
`=1/2(V(0)-V(n))`
`=1/2(1-1/(2n+1))`
`=n/(2n+1)`
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