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Homework Help: Series Question

  1. Jan 8, 2010 #1
    If n is a positive integer, then the limit of [tex]\sum k/n^2[/tex] from k=1 to n as n approaches infinity = ?

    A) pi
    B) 1
    C) 1/2
    D) 1/pi
    E) 0

    How do I do this?
     
  2. jcsd
  3. Jan 8, 2010 #2

    dextercioby

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    Can you compute the sum ?
     
  4. Jan 8, 2010 #3
    c) 1/2

    since 1/n^2 is constant remove it outside
    sum of n natural numbers formula can be used for the summation
    then you have lim tending to infinity
    n(n+1)/(2n^2)
    use L 'opital

    should be right
     
  5. Jan 8, 2010 #4

    LCKurtz

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    Except that, technically, L'Hospital's rule does not apply since n is given to have integer values. Just divide the numerator and denominator by n2 and use basic properties of limits.
     
  6. Jan 9, 2010 #5

    HallsofIvy

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    If a sequence of functions, f(x), has limit L as x goes to a, then any sequence, {f(xn)}, with {xn} converging to a, must also converge to L. In particular, if f(x) goes to L as x goes to infinity, the sequence {f(n)} also converges to L. As long as a function of n, for n a positive integer, can be as a function of x, a real variable, (for example, does NOT involve factorials), L'Hopital's rule can be applied.
     
  7. Jan 9, 2010 #6

    LCKurtz

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    Yes, of course. That's why I said "technically". Still, I think it is best if students learn to use the more elementary methods when they are appropriate.
     
  8. Jan 9, 2010 #7
    personally i wouldnt prefer L opital either
     
  9. Jan 10, 2010 #8
    Thanks, the answer is C. But, is there another way to do this?
     
  10. Jan 10, 2010 #9

    LCKurtz

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    You have been given two methods. Did you understand both? An experienced person doing a multiple choice question where you didn't have to show work or give a reason would have would just have looked at the ratio of the n2 terms to get the answer.
     
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