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Show tha , lim(ntooo) ((1)/( n +1)+(1)/...

Show tha , `lim_(ntooo) ((1)/( n +1)+(1)/( n +2)+ . . . +(1)/( 6 n))= log 6`.

Text Solution

Verified by Experts

`underset(ntooo)(lim) ((1)/(n+1)+(1)/(n+2)+. . . + (1)/(6n))=sum _(r=1)^(5n)(1)/(n+r)`
`=underset(n tooo)(lim) (1)/(n)sum_(r=1)^(5n)(1)/((1+(r)/(n)))`
`=int_(0)^(5)(dx)/(1+x)=[log(1+x)]_(0)^(5)=log6 - log1=6`
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