Evaluate: `(lim)_(nvecoo)([(n+1)(n+2)(n+n)^(1/n))/n`

Explanation

Let `L=lim_(ntooo)([(n+1)(n+2)……..(n+n)]^(1/n))/n`
`=lim_(ntooo)[((n+1)(n+2)……….(n+n))/(n^(n))]^(1//n)`
`implies lim_(ntooo)[(n+1)/n . (n+2)/n……….(n+n)/n]^(1//n)`
`=lim_(ntooo) [(1+1/n)(1+2/n)……..(1+n/n)]^(1//n)`
`:. Log L=lim_(ntooo) 1/n[log(1+1/n)+log(1+2/n)+............+log(1+n/n)]`
`=lim_(ntooo) sum_(r=1)^(n) 1/nlog(1+r/n)=int_(0)^(1)log(1+x)dx`
`[xlog(1+x)]_(0)^(1)-int_(0)^(1)x/(1+x)dx`
`=log2-int_(0)^(1)[1-(1//(1+x))]dx`
`=log2-[x-log(1+x)]_(0)^(1)`
`=log2-[(1-log2)-(0-log1)]`
`2log2-1=log(2^(2)//e)`
`:.L=2^(2)//3=4//e`