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lim(n->oo) (1^p+2^p+3^p+...........+n^p)...

`lim_(n->oo) (1^p+2^p+3^p+...........+n^p)/n^(p+1)`

A

`1/(p+1)`

B

`1/(1-p)`

C

` 1/p - 1/(p-1)`

D

`1/(p+2)`

Text Solution

Verified by Experts

The correct Answer is:
A
Given , `lim_(ntoinfty)(1^(p)+2^(p)+3^(p)+...+n^(p))/(n^(p+1))=lim_(ntoinfty)sum_(r=1)^(n)[(r^(p))/(n^(p+1))]`
`=lim_(ntoinfty)(1)/(n)sum_(r=1)^(n)((r)/(n))^(p)=int_(0)^(1)x^(p)dx=[(x^(p+1))/(p+1)]_(0)^(1)=(1)/(p+1)`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-PRACTICE EXERCISE (Exercise 2) (MISCELLANEOUS PROBLEMS)
  1. If kint(0)^(1)xf(3x)dx=int(0)^(3)tf(t)dt, then the value of k is

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  2. lim(nto oo)+(1)/(sqrt(n^(2)+n))+(1)/(sqrt(n^(2)+2n))+...(1)/(sqrt(n^(2...

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  3. lim(n->oo) (1^p+2^p+3^p+...........+n^p)/n^(p+1)

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

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  5. If f(x) = tanx-tan ^(3) x + tan^(5) x - tan ^(7) x + ... infty for o...

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  6. The value of int(0)^((pi)/(8))cos^(3)4 theta d theta is equal to -

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  7. int (0)^(pi//3) (cos x + sin x)/(sqrt(1+sin 2x))dx is equal to

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  8. int (0)^(1) x^(3//2) sqrt(1-x) dx is equal to

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  9. The value of int (1)^(2) (dx)/((x+1) sqrt((x^2)-1)) is

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  10. If f(x)=|(sin x+sin x 2x+sin3x,sin2x,sin3x),(3+4sinx,3,4sinx),(1+sinx,...

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  11. int (-1)^(1) (e^(x^(3)) +e^(-x^(3))) (e^(x)-e^(-x)) dx is equal to

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  12. underset(n to oo)lim((1)/(n)+(1)/(n+1)+...+(1)/(3n)) is equal to

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  13. Let f (x) = x – [x], for every real number x, where [x] is the greates...

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  14. The value of int (0)^(1) (x^(4) +1)/(x^(2)+1)dx is

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  15. The value of overset(5)underset(3)int (x^(2))/(x^(2)-4)dx, is

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  16. The value of int (0)^(pi//2) ((sin x + cos x)^(2))/(sqrt(1+sin 2x) dx...

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  17. int (0)^(a) sqrt(a^(2) - x^(2)) dx is equal to

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  18. The value of int(e^(-1))^(e) (dt)/(t(t+1)) is equal to

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  19. The value of int (1)^(e) 10^(log(e)x) dx is equal to

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  20. If int (2)^(e) (1/(logx)-1/(logx)^(2))dx = a + b/(log2) , then

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