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Find the sum Sigma(r=1)^(oo) (r-2)/((r+...

Find the sum `Sigma_(r=1)^(oo) (r-2)/((r+2)(r+3)(r+4))`

Text Solution

Verified by Experts

The correct Answer is:
`1/12`
`T_(r)=(r-2)/((r+2)(r+3)(r+4))=(r(r+4)-(r+1)(r+2))/((r+2)(r+3)(r+4))`
`=r/((r+2)(r+3))-((r+1))/((r+3)(r+4))`
`therefore S_(n)=sum_(r=1)^(n)T_(r)=1/12-((n+1))/((n+3)(n+4))`
`thereforeS_(oo)=1/12`
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