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Lt(n rarr oo) sum(r=1)^(n)[(1)/(sqrt(4n^...

`Lt_(n rarr oo) sum_(r=1)^(n)[(1)/(sqrt(4n^(2) - r^(2)))]`

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The correct Answer is:
`(pi)/(6)`
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Find Lt(n rarr oo) sum_(r=0)^(n-1)(1)/(sqrt(n^(2) - r^(2))

{:(" "Lt),(n rarr oo):}1/n sum_(r=1)^(2n)(r)/(sqrt(n^(2)+r^(2)))=

Lt_(ntooo)sum_(r=1)^(n)(1)/(sqrt(4n^(2)-r^(2)))=

Lt_(ntooo)sum_(r=1)^(n-1)(1)/(sqrt(n^(2)-r^(2)))=

Lt_(n rarr oo)sum_(r=0)^(n-1)((r)/(n^(2) + r^(2)))

Lt_(ntooo)(1)/(n)sum_(r=1)^(2n)(r)/(sqrt(n^(2)+r^(2)))=

{:(" " Lt),(n rarroo):} sum_(r=1)^(n)(r)/(n^(2)+r^(2))=

Lt_(n rarr oo)[(1)/(n)+(1)/(sqrt(n^(2) -1^(2)))+(1)/(sqrt(n^(2)-2^(2)))+... "to n terms"]

Evaluate the following define integrals as limit of sums : lim_(n rarr oo) sum_(i=1)^(n) (i)/(n^(2)+i^(2))

AAKASH SERIES-DEFINITE INTEGRALS-Exercise - 2.7 (Level-2)
  1. Evaluate the limit . underset(n to 00)("lim") [(1)/(n+1)+(1)/(n+2)+…...

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  2. {:(" "Lt),(n rarr oo):}{ (1)/(2n+1)+(1)/(2n+2)+......+(1)/(3n)}=

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  3. Lt(n rarr oo)[(1)/(3n+1) + (1)/(3n+2)+…+(1)/(4n)]

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  4. Lt(n rarr oo)sum(r=0)^(n-1)((r)/(n^(2) + r^(2)))

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  5. Lt(ntooo)sum(r=1)^(n)(1)/(n)[sqrt ((n+r)/(n-r))]

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  6. Lt(n rarr oo) sum(r=1)^(n)[(1)/(sqrt(4n^(2) - r^(2)))]

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  7. Evaluate the limit. underset(n to 00)("lim") (sqrt(n+1)+sqrt(n+2)+…...

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  8. Lt(ntooo){(1)/(sqrt(n^(2)+1))+(1)/(sqrt(n^(2)+2^(2)))+.......+(1)/(sqr...

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  9. Lt(n rarr oo)[(1)/(n)+(1)/(sqrt(n^(2) -1^(2)))+(1)/(sqrt(n^(2)-2^(2)))...

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  10. Lt(n rarr oo)[(1n^(2))/((n+1)^(3))+(n^(2))/((n+2)^(3))+(n^(2))/((n+3))...

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  11. {:(" "Lt),(n rarr oo):} [(sqrt(n^(2)-1^(2)))/(n^(2))+(sqrt(n^(2)-2^(2)...

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  12. Lt(ntooo)(1)/(n)[sec^(2)""(pi)/(4n)+sec^(2)""(2pi)/(4n)+......+sec^(2)...

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  13. Lt(n rarr oo)[(n^(2))/((n^(2) +1)^(3//2))+(n^(2))/((n^(2) + 2)^(3//2))...

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  14. Lt(n rarr oo)[(sqrt(n))/(sqrt(n^(3)))+(sqrt(n))/(sqrt((n+4)^(3)))+.......

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  15. {:(" "Lt),(n rarr oo):} [ (1)/(n^(2)) sec^(2). (1)/(n^(2)) + (2)/(n^(2...

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  16. Lt(n rarr oo)[(1^(3))/(n^(4)+1^(4))+(2^(3))/(n^(4) + 2^(4))+....+(n^(3...

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  17. Lt(n rarr oo)(1)/(n)[e^((1)/(n))+e^((2)/(n))+....+e^((n)/(2))]

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  18. Lt(n rarr oo)[(1^(99)+2^(99)+3^(99)+...+n^(99))/(n^(100))]

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  19. Evaluate underset(n to oo)("lim")[(1+(1)/(n))(1+2/n)* * * (1+(n)/(n))...

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  20. Lt(n rarr oo)[(1+(1)/(n^(2)))^((2)/(n^(2)))(1+(2^(2))/(n^(2)))^((4)/(n...

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