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Expressing the limit of a sum as a definite integral

  1. Nov 30, 2012 #1
    1. The problem statement, all variables and given/known data
    Express the following as a definite integral:

    Express the attached limit as an integral.


    3. The attempt at a solution
    I have gotten as far as every part of the answer except the upper bound. the answer is:
    10
    (from 1 to 10) [x-4lnx]dx
    1

    since the definition of the definite integral is:
    a
    f(x)dx = lim Ʃ Δxif(x)
    b________Δ→∞ i=1

    i set Δxi = 9/n since that approaches zero. f(x) would be left to 1+9i/n - 4ln(1+9i/n)
    so i set x = 1+9i/n.
    since n approaches ∞ and the upper bound of the sum is ∞, i plugged ∞ in for i and n.
    thats where I have trouble. ∞/∞ is undefined. when i plug 1 in i end up with 1 so that is the lower bound.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

  2. jcsd
  3. Nov 30, 2012 #2

    mfb

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    2016 Award

    Staff: Mentor

    I don't think your infinite sum converges for any i - you sum over i which grows like i^2 and the log-expression does not reduce this enough (just grows with i*log(i)).
    If the sum is supposed to run from i=1 to n, this makes sense, and you get the maximal x-value simply by setting i=n.
     
  4. Nov 30, 2012 #3

    Dick

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    Science Advisor
    Homework Helper

    There's a typo in the attached limit expression. The upper limit on the summation should be n. As written it doesn't approach anything. The sum by itself diverges.
     
    Last edited: Nov 30, 2012
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