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Find the sum of the series sum(r=11)^(99...

Find the sum of the series `sum_(r=11)^(99)(1/(rsqrt(r+1)+(r+1)sqrtr))`

Text Solution

Verified by Experts

The correct Answer is:
9/10
`T_(r)=1/(sqrtrsqrt(r+1)[sqrtr+sqrt(r+1)])=(sqrt(r+1)-sqrtr)/(sqrtrsqrt(r+1))`
`=1/sqrtr-1/(sqrt(r+1))`
=V( r)-V(r+1), where V(r )=`1/(sqrtr)`
`therefore` Required sum, `sum_(r=1)^(99)(V(r )-V(r+1))=V(1)-V(100)`
`=1-1/(sqrt(100))`
`=1-1/10=9/10`
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