Home
Class 12
MATHS
lim(n -> oo) (((n+1)(n+2)(n+3).......3n)...

`lim_(n -> oo) (((n+1)(n+2)(n+3).......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))`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE|Exercise JEE Advanced Previous Year|38 Videos

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise (Numerical)|28 Videos

  • CURVE TRACING

    CENGAGE|Exercise Exercise|24 Videos

  • DETERMINANT

    CENGAGE|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

The value of lim_(n->oo) [tan(pi/(2n)) tan((2pi)/(2n))........tan((npi)/(2n))]^(1/n) is

Evaluate: lim_(n->oo)(((n+1)(n+2)...(n+n))^(1/n))/n

lim_(n->oo)((n^2-n+1)/(n^2-n-1))^(n(n-1)) is

lim_(n to oo)((1)/(n^2)+(2)/(n^2)+(3)/(n^2)+"………."+(n)/(n^2)) is

lim_(n->oo)(n(2n+1)^2)/((n+2)(n^2+3n-1)) is equal to (a)0 (b) 2 (c) 4 (d) oo

The value of lim_(n rarroo) sum_(r=1)^(n)(1)/(sin{((n+r)pi)/(4n)}).(pi)/(n) is equal to

lim_(xto oo)((1)/(1-n^(2))+(2)/(1-n^(2))+ . . .+(n)/(1-n^(2))) is

lim_(ntoinfty) (n/(n^2+1^2)+n/(n^2+2^2)+n/(n^2+3^2)+...+1/(2n)) is equal to

lim_(nrarroo) [(1)/(n)+(n^(2))/((n+1)^(3))+(n^(2))/((n+2)^(3))+...+(1)/(8n)] is equal to

lim_(n rarr 0)(1/(1+n^(2))+2/(1+n^(2))+3/(1+n^(2))+....+n/(1+n^(2))) is equal to