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Evalute lim(n->oo)[1/((n+1)(n+2))+1/((n+...

Evalute `lim_(n->oo)[1/((n+1)(n+2))+1/((n+2)(n+4))+......+1/(6n^2)]`

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Verified by Experts

The given limit is
`L=lim_(ntooo)sum_(r=1)^(n)n . 1/((n+r)(n+2r))`
`=lim_(ntooo)1/n sum_(r=1)^(n)1/((1+r//n)(1+2r//n))`
`=int_(0)^(1)(dx)/((1+x)(1+2x))`
`=int_(0)^(1)((-1)/(1+x)+2/(1+2x))dx`
`=[-log(1+x)+log(1+2x)]_(0)^(1)`
`=[(-log2+log3)-(-log1+log1)]`
`=log(3//2)`
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CENGAGE ENGLISH-DEFINITE INTEGRATION -JEE ADVANCED
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