Home
Class 12
MATHS
Evaluate lim(ntooo) n^(-n^(2))[(n+2^(0))...

Evaluate `lim_(ntooo) n^(-n^(2))[(n+2^(0))(n+2^(-1))(n+2^(-2))...(n+2^(-n+1))]^(n)`.

Text Solution

Verified by Experts

We have `underset(ntooo)limn^(-n^(2))[(n+1)(n+(1)/(2))...(n+(1)/(2^(n-1)))]^(n)`
`=underset(ntooo)lim[((n+1)(n+(1)/(2))...(n+(1)/(2^(n-1))))/(n^(n))]^(n)`
`=underset(ntooo)lim((n+1)/(n))^(n).((n+(1)/(2))/(n))^(n)...((n+(1)/(2^(n-1)))/(n))^(n)`
`=underset(ntooo)lim(1+(1)/(n))^(n).(1+(1)/(2n))^((2n)/(2))...(1+(1)/(2^(n-1)n))^((2^(n-1).n)/(2^(n-1)))`
`=underset(ntooo)lime^(1).e^((1)/(2)).e^(2^((1)/(2)))...e^((1)/(2^(n-1)))`
`=e^(1).e^((1)/(2)).e^(2^((1)/(2)))...("infinite factors")`
`=e^((1)/(1-(1)/(2)))=e^(2)`
Promotional Banner

Topper's Solved these Questions

  • LIMITS

    CENGAGE PUBLICATION|Exercise EXERCISE 2.1|10 Videos

  • LIMITS

    CENGAGE PUBLICATION|Exercise EXERCISE 2.2|7 Videos

  • LIMITS

    CENGAGE PUBLICATION|Exercise Comprehension Type|4 Videos

  • JEE 2019

    CENGAGE PUBLICATION|Exercise Chapter 10|8 Videos

  • LINEAR COMBINATION OF VECTORS, DEPENDENT AND INDEPENDENT VECTORS

    CENGAGE PUBLICATION|Exercise DPP 1.2|10 Videos

Similar Questions

Explore conceptually related problems

Evaluate lim_(ntooo) (1)/(1+n^(2))+(2)/(2+n^(2))+...+(n)/(n+n^(2)).

Evaluate lim_(ntooo) nsin(2pisqrt(1+n^(2))),(n inN).

Evaluate lim_(ntooo) sin^(n)((2pin)/(3n+1)),ninN.

Evaluate lim_(ntooo) (1)/(n^(2(log_(e)n-log_(e)(n+1)))+n) .

Evaluate lim_(ntooo) (1^(3)+2^(3)+3^(3)+...+n^(3))/(sqrt(4n^(8)+1)).

Prove that, lim_(ntooo)[(n)/(n^(2))+(n+1)/(n^(2))+(n+2)/(n^(2))+.......+(2n)/(n^(2))]=(3)/(2)

Evaluate lim_(ntooo) (n^(p)sin^(2)(n!))/(n+1),"where "0ltplt1.

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

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

Evalute : lim_(ntooo)[(n)/(n^2+1^2)+(n)/(n^2+2^2)+........+(1)/(2n)] .