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Evaluate ("lim")(n rarr oo)sum(k=1)^nk/(...

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

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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`
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CENGAGE PUBLICATION-DEFINITE INTEGRATION -CAE_TYPE
  1. Evaluate: ("lim")(n rarr oo)(1/(sqrt(4n^2-1))+1/(sqrt(4n^2-2^2))+ .......

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  2. Evaluate: underset(nrarrinfty)lim[(1/n^2sec^2(1/(n^2))+2/n^2sec^2((4)/...

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  3. Evaluate ("lim")(n rarr oo)sum(k=1)^nk/(n^2+k^2)

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  4. Evaluate the following limit: lim(nto oo)(sum(r=1)^(n) sqrt(r)sum(r=...

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  5. Evaluate the following limit: lim(nto oo)[(n!)/(n^(n))]^(1//n)

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  6. Prove that 4 lt= int1^3 sqrt(3+x^2) dx lt= 4sqrt(3)

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  7. If I1=int0^1 2^(x^2)dx , I2=int0^1 2^(x^3) dx , I3= int1^2 2^(x^2) dx ...

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  8. If II=int0^(pi//2)cos(sinx)dx ,I2=int0^(pi/2)sin(cosx)dx \ ,a n d \ I3...

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  9. Show that : pi/6leint0^1dx/sqrt(4-x^2-x^3)lepi/(4sqrt2)

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  10. Evaluate int(0)^(pi//2)|sinx-cosx|dx.

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  11. Evaluate: int(-1)^4f(x)dx=4a n dint2^4(3-f(x))dx=7, then find the val...

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  12. Evaluate int(1)^(5)sqrt((x-2)sqrt(x-1))dx.

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  13. Evaluate: int(-1)^3(tan^(-1)x/(x^2+1)+tan^(-1)(x^2+1)/x)dx

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  14. Evaluate int(1)^(a)x.a^(-[log(a)x])dx,(agt1).Here [.] represents the g...

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  15. Evaluate: int1^(e^6)[(logx)/3]dx ,w h e r e[dot] denotes the greatest...

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  16. Find the value of int(-1)^1[x^2+{x}]dx ,w h e r e[dot]a n d{dot} denot...

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  17. Prove that int0^x[cot^(-1)x]dx ,w h e r e[dot] denotes the greatest in...

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  18. Prove that int0^xf[t]dt=([x]([x]-1))/2+[x](x-[x]), where [.] denotes...

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  19. Evaluate: int0^oo[2e^(-e)]dx ,w h e r e[x] represents greatest intege...

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  20. If f(a+b-x)=f(x), then prove that inta^b xf(x)dx=(a+b)/2inta^bf(x)...

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