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

Find the sum of infinite series
`(1)/(1xx3xx5)+(1)/(3xx5xx7)+(1)/(5xx7xx9)+….`

Text Solution

Verified by Experts

The correct Answer is:
`1/(12)`
`T_(r)=1/((2r-1)(2r+1)(2r+3))`
`=1/4cdot((2r+3)-(2r-1))/((2r-1)(2r+1)(2r+3))`
`=1/4[1/((2r-1)(2r+1))-1/((2r+1)(2r+3))]`
`=1/4[V(r-1)-V(r )],` where V( r)=`1/((2r+1)(2r+3))`
`thereforesum_(r=1)^(n)T_(r)=sum_(r=1)^(n)1/4[V(r-1)-V(r )]`
`=1/4[V(0)-V(n)]`
`=1/4[1/3-1/((2n+1)(2n+3))]`
`therefore` Sum of infinite terms=`1/4[1/3-0]=1/12`
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