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lim(n to oo)((1)/(n^2)+(2)/(n^2)+(3)/(n^...

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

A

`1//4`

B

`1//6`

C

`1//9`

D

`1//12`

Text Solution

Verified by Experts

The correct Answer is:
D
`f(x)=lim_(nto oo)( 1/n)(x+1/n)^(2)+(x+2/n)^(2)+………+(x+(n-1)/n)^(2)`
`=int_(0)^(1)(x+y)^(2)dy`
`=x^(2)+x+1/3`
`:. f(x)=(x+1/2)^(2)+1/3-1/4`
`:.f(x)|_("min")=1/12`
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CENGAGE-DEFINITE INTEGRATION -Exercise (Single)
  1. lim(n to oo)((1)/(n^2)+(2)/(n^2)+(3)/(n^2)+"………."+(n)/(n^2)) is

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  2. IfSn=[1/(1+sqrt(n))+1/(2+sqrt(2n))++1/(n+sqrt(n^2))],t h e n(lim)(nvec...

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  3. The value of ("lim")(nvecoo)sum(r=1)^(4n)(sqrt(n))/(sqrt(r)(3sqrt(r)+s...

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  4. lim(xto oo)(1^(2)+2^(2)+ . .. +n^(2))/(n^3)

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  5. T h ev a l u eof(lim)(nvecoo)[t a npi/(2n)tan(2pi)/(2n)dottan(npi)/(2n...

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  6. int(2-a)^(2+a)f(x)dxi se q u a lto[w h e r ef(2-alpha)=f(2+alpha)AAalp...

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  7. about to only mathematics

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  8. Which of the following is incorrect ?

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  9. int(-1)^(1/2)(e^x(2-x^2)dx)/((1-x)sqrt(1-x^2))i se q u a lto (sqrt(e)...

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  10. Ifint(log2)^x(dx)/(sqrt(e^x-1))=pi/6,"then " x " is equalto" (a)4 (b)...

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  11. int(5/2)^5(sqrt((25-x^2)^3))/(x^4)dx is equal to (a) pi/6 (b) (2pi...

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  12. If f(x) is integrable over [1,2] then int(1)^(2)f(x)dx is equal to

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  13. int(e^x)/(e^(2x)+4)dx

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  14. The value of the integral int0^1(dx)/(x^2+2xcosalpha+1) is equal to si...

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  15. int0^oo(dx)/([x+sqrt(x^2+1)]^3)i se q u a lto 3/8 (b) 1/8 (c) -3/8 ...

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  16. Iff(y)=e^y ,g(y)=y;y>0,a n dF(t)=int0^t f(t-y)g(y) dy,t h e n a) F(t...

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  17. Let p(x) be a real polynomial of least degree which has a local maximu...

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  18. The numbers of possible continuous f(x) defined in [0,1] for which I1...

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  19. Suppose that F(x) is an anti-derivative of f(x)=(sinx)/x ,w h e r ex >...

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  20. int(-pi/3)^0[cot^(-1)(2/(2cosx-1))+cot^(-1)(cosx-1/2)]dx i s equal to...

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