`lim_(n -> oo) (((n+1)(n+2)(n+3).......3n) / n^(2n))^(1/n)`is equal toA. `27/(e^(2)0`B. `9/(e^(2))`C. `3log3-2`D. `18/e^(4)`

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

1.	`lim_(n -> oo) (((n+1)(n+2)(n+3).......3n) / n^(2n))^(1/n)`is equal toA. `27/(e^(2)0`B. `9/(e^(2))`C. `3log3-2`D. `18/e^(4)`
Answer» Correct Answer - 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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Given that `lim_(nrarroo) sum_(r=1)^(n) (log_(e)(n^(2)+r^(2))-2log_(e)n)/n = log_(e)2+pi/2-2`, then evaluate : ` lim_(nrarroo) (1)/(n^(2m))[(n^(2)+1^(2))^(m)(n^(2)+2^(2))^(m) "......"(2n^(2))^(m)]^(1//n)`.
`lim_(nrarroo) sum_(r=0)^(n-1) (1)/(sqrt(n^(2)-r^(2)))`
`lim_(nrarroo) (((n+1)(n+2)"........"3n)/(n^(2n)))^(1//n)` is equal to :A. `(27)/(e^(2))`B. `(9)/(e^(2))`C. `3 log3-2`D. `(18)/(e^(4))`