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lim(n -> oo) (((n+1)(n+2)(n+3).......2n)...

`lim_(n -> oo) (((n+1)(n+2)(n+3).......2n) / n^(2n))^(1/n)`is equal to

Text Solution

Verified by Experts

The correct Answer is:
`1.50`
`lim_(nrarroo)n^(-n^(2))((n+1)(n+(1)/(3))(n+(1)/(3^(2)))"............"(n+(1)/(3^(n-1))))^(n)`
`=lim_(nrarroo)((n+1)/(n))^(n).((n+(1)/(3))/(n))^(n)"............"((n+(1)/(3^(n-1)))/(n))^(n)=lim_(nrarroo)(1+(1)/(n))^(n)(1+(1)/(3n))^((3n)/(3))"...."(1+(1)/(3^(n-1).n))^((3^(n-1).n)/(3^(n-1)))`
`e.e^((1)/(3)).e^((1)/(9))"........"e^((1)/(3^(n-1)))=e^(1+(1)/(3)+(1)/(3^(2))"......."oo)=e^((1)/(1-1//3))=e^(3//2)" , "k=1.50`
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