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Calculus 2 - Infinite Series Question - Estimating Series with Positive Terms

  1. Oct 21, 2011 #1
    1. The problem statement, all variables and given/known data

    Consider the following convergent series. Then complete parts a throw d below.

    sum[k=1,inf] 5/k^7

    a. Find an upper bound for the remainder in terms of n

    2. Relevant equations

    Estimating Series with Positive Terms
    Let f be a continuous, positive, decreasing function for x >= 1 and let a_k = f(k) for k = 1,2,3,.... Let S = sum[k=1,inf] a_k be a convergent series and let S_n = sum[k=1,n] a_k be the sum of the first n terms of the series. The remainder R_n = S - S_n satisfies

    R_n <= integral[n,inf] f(x)dx.

    Furthermore, the exact value of the series is bounded as follows:

    S_n + integral[n+1,inf] f(x)dx <= sum[k=1,inf] a_k <= S_n + integral[n,inf] f(X)dx

    3. The attempt at a solution

    I'm unsure how to do this problem. I believe that I'm trying to evaluate

    S_n + integral[n,inf] f(X)dx

    I have no problem find the value of integral[n,inf] f(X)dx
    but am not sure how to find the value of S_n. I would now how to find the value of this if I was asked to find upper bound for the error for the first 50 terms, I could then find S_50 by just finding the sum which would be a finite number, but I am unsure how to find the upper bound in this case were I guess I'm trying to find the value of S_n in this case would be S_inf which I'm not sure how to do. Thank's for any help which you can provide me with.
     
  2. jcsd
  3. Oct 21, 2011 #2

    Mark44

    Staff: Mentor

    The remainder (or error) is
    [tex]\int_n^{\infty}f(x)dx = \int_n^{\infty}\frac{dx}{x^7} [/tex]
     
  4. Oct 22, 2011 #3
    Well this problem was one of my homework questions which I do online in this program in which I input my answer and it told me I was wrong when I entered 5/6. Have I done something wrong?

    5*integral[1,inf] dk/k^7 = 5/6
     
  5. Oct 22, 2011 #4

    Mark44

    Staff: Mentor

    Yes. If you want to estimate the series by using the first 50 terms of the series, the error is
    [tex]R_{50} = \int_{50}^{\infty}5x^{-7}dx[/tex]

    You have this in your relevant equations, but you must not have thought it to be relevant...
     
    Last edited: Oct 22, 2011
  6. Oct 22, 2011 #5
    Alright well I entered 5/(6n) and it still told me I was wrong.
     
  7. Oct 22, 2011 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You have been told that an upper bound for the error is
    [tex]\int_n^\infty \frac{5}{x^7}dx= 5\int_n^\infty x^{-7}dx[/tex]

    What is that? (It is NOT 5/(6n)!)
     
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