Home
Class 12
MATHS
Evaluate ("lim")(nvecoo)sum(k=1)^nk/(n^2...

Evaluate `("lim")_(nvecoo)sum_(k=1)^nk/(n^2+k^2)`

Text Solution

Verified by Experts

The correct Answer is:
`1/2log2`
`lim_(nto oo) sum_(K=1)^(n)K/(n^(2)+K^(2))`
`=lim_(nto oo) sum_(K=1)^(n) 1/(n^(2))xx K/(1+(K/n)^(2))`
`=lim_(nto oo) sum_(K=1)^(n)1/n xx(K//n)/(1+(K/n)^(2))`
`=lim_(n to oo) sum_(K=1)^(n)1/nxx(K//n)/(1+(K/n)^(2))`
`=int_(0)^(1)x/(1+x^(2))dx`
`=1/2 log (1+x^(2))|_(0)^(1)=1/2log2`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise 8.3|4 Videos

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise 8.4|10 Videos

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise 8.1|4 Videos

  • CURVE TRACING

    CENGAGE|Exercise Exercise|24 Videos

  • DETERMINANT

    CENGAGE|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

Evaluate: ("lim")_(nvecoo)[1/(n^2)sec^2 1/(n^2)+2//n^2sec^2 4/(n^2)++1/nsec^2 1]

The value of ("lim")_(nvecoo)sum_(r=1)^(4n)(sqrt(n))/(sqrt(r)(3sqrt(r)+sqrt(n))^2) is equal to 1/(35) (b) 1/4 (c) 1/(10) (d) 1/5

If f(x) is integrable over [1,], then int_1^2f(x)dx is equal to ("lim")_(nvecoo)1/nsum_(r=1)^nf(r/n) ("lim")_(nvecoo)1/nsum_(r=n+1)^(2n)f(r/n) ("lim")_(nvecoo)1/nsum_(r=1)^nf((r+n)/n) ("lim")_(nvecoo)1/nsum_(r=1)^(2n)f(r/n)

Evaluate: ("lim")_(nvecoo)(1/(sqrt(4n^2-1))+1/(sqrt(4n^2-2^2))++1/(sqrt(3n^2)))

Evaluate: ("lim")_(nvecoo)((1^2+2^2+3^3++n^2)(1^3+2^3+3^3++n^3))/(1^6+2^6+3^6++n^6)

Evaluate: ("lim")_(nvecoo)(sumr=1nsqrt(r)sumr=1n1/(sqrt(r)))/(sumr=1n r)

Evaluate: ("lim")_(nvecoo)n^(-n^2){(n+2^0)(n+2^(-1))(n+2^(-2))(n+2^(-n+1))}^n

Evaluate lim_(n->oo) [sum_(r=1)^n(1/2) ^r] , where [.] denotes the greatest integer function.

Find the value of lim_(n rarr oo)sum_(k=1)^(n)(k^(2)+k-1)/((k+1)!) .

Find the value of sum_(p=1)^n(sum_(m=p)^n^n C_m^m C_p)dot And hence, find the value of (lim)_(nvecoo)1/(3^n)sum_(p=1)^n(sum_(m=p)^n^n C_m^m C_p)dot