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

Answer

1

Explanation

`y_(n)=[((n+1)(n+2)………(n+n))/(n^(n))]^(1//n)`
`impliesy_(n)=((n+1)/n . (n+2)/n . ………… .(n+n)/n)^(1/n)`
`:. logy_(n)=log((n+1)/n . (n+2)/n. ………… . (n+n)/n)^(1/n)`
`=1/nsum_(r=1)^(n)"log"(n+r)/n`
`=1/nsum_(r=1)^(n)log(1+r/n)`
`:. log_(nto oo) logy_(n)=lim_(n to oo) 1/n sum_(r=1)^(n)log(1+r/n)`
`:.log(lim_(nto oo) y_(n))=int_(0)^(1)log(1+x)dx`
`:.logL=int_(1)^(2)logx dx`
`=|xlogx-x|_(1)^(2)-2log2-1`
`="log"4/e`
`implies L=4/e`
`implies[L]=1`