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underset(nrarrinfty)lim((n+1)(n+2)....3n...

`underset(nrarrinfty)lim((n+1)(n+2)....3n)/(n^(2n)))^(1/n)` is equal to

A

`27/(e^(2)0`

B

`9/(e^(2))`

C

`3log3-2`

D

`18/e^(4)`

Text Solution

Verified by Experts

The correct Answer is:
A
`L=int_(nto oo) (((n+1)(n+2)……….(n+2n))/(n^(2n)))^(1//n)`
`:. log_(e)L=1/n(lim_(nto oo) sum_(r=1)^(2n)log(1+4/n))`
`:.log_(e)L=int_(0)^(2)log(1+x)dx`
`:.log_(e)L(xlog(1+x))_(0)^(2)-int_(0)^(2)x/(1+x)dx`
`:.log_(e)L=2log_(e)3-int_(0)^(2)(1-1/(1+x))dx`
`:. log_(e)L=2log3-(x-log(1+x))_(0)^(2)`
`=log_(e)L=2log3-(2-log3)`
`:.log_(e)L=3log3-2="log"27/(e^(2))`
`:.L=27/(e^(2))`
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