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Exact sum of a series

  1. Sep 22, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the exact sum of:

    [itex]\sum[/itex] 2/(n7n)

    n=1->∞

    3. The attempt at a solution

    Let Sn denote the nth partial sum.

    ln(Sn) = [itex]\Sigma[/itex] ln(2/(n7n))

    =[itex]\Sigma[/itex] ln2 - lnn - nln7

    = nln2 - (ln1 + ln7 + ln2 + 2ln7 + ln3 + 3ln7 ...)
    = nln2 - ln(n!) - ln7[itex]\Sigma[/itex] n
    = ln (2n)/n! - ln7 ([itex]\Sigma[/itex] n)

    I'm not sure where to go from here, or if I'm even going in the right direction.
     
  2. jcsd
  3. Sep 22, 2012 #2

    jbunniii

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    Do you know how these two power series relate to each other?
    [tex]\sum z^{n-1}[/tex]
    [tex]\sum \frac{1}{n} z^n[/tex]
     
  4. Sep 22, 2012 #3
    No, but now I intend to find out. A hint would be appreciated though.
     
  5. Sep 22, 2012 #4
    Never mind. Those two series are related by a factor of 1/zn. I should have looked a bit closer :/.
     
  6. Sep 22, 2012 #5

    jbunniii

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    Hint: [itex]\int z^{n-1} dz = \frac{1}{n} z^n[/itex]
     
  7. Sep 22, 2012 #6

    vela

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    This approach won't work because the very first step is wrong. It's not true that ln(a+b) = ln a + ln b.
     
  8. Sep 22, 2012 #7

    Ray Vickson

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    Stop right there. What you have done is invalid: the log of a sum is not the sum of the logs. For example, if you claim that log(2 + 3) = log(2) + log(3) you would have log(2) + log(3) = log(2*3) = 6 (because the log of a product is the sum of the logs), so your method would give log(5) = log(6), hence 5 = 6.

    RGV
     
  9. Sep 22, 2012 #8
    Thanks for the correction, I normally know log rules, just that one slipped. You guys mean bringing the log into the sum, right?
     
  10. Sep 22, 2012 #9
    So I just evaluate the integral?
     
  11. Sep 22, 2012 #10

    jbunniii

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    Yes, do you see how this can be applied to this problem?
     
  12. Sep 22, 2012 #11

    Ray Vickson

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    I thought that was what I said.

    RGV
     
  13. Sep 22, 2012 #12
    Yes, I understand now, thank you very much!
     
  14. Sep 22, 2012 #13
    Hey I was just wondering how you can take the integral of a series to find the sum? Don't series only include integers and will not add up to the same thing as a integral? For example my calculus textbook says when doing the integral test specifically states that the integral is not the sum of the series. Can someone please tell me what im missing? Thank you.
     
  15. Sep 22, 2012 #14

    jbunniii

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    If
    [tex]f(z) = \sum a_n z^n[/tex]
    then, under certain conditions, you can perform manipulations like this:
    [tex]\int f(z) dz = \int \sum a_n z^n dz = \sum \int a_n z^n dz = \sum \frac{a_n}{n+1} z^{n+1} dz[/tex]
    You need to provide justification for interchanging the order of integration and summation. Uniform convergence of the power series is a sufficient condition.

    This kind of trick is useful for finding sums such as [itex]\sum (a_n / n) z^n[/itex], provided you have a closed-form expression for [itex]\sum a_n z^n[/itex]. Term by term differentiation of the power series can also be useful in some situations.
     
  16. Sep 23, 2012 #15

    SammyS

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    The idea here is to recognize that some fairly well-known power series in x (or z as jbunniii seems to like) is involved here, and that this power series represents a function of x. Then plug-in a value for x which gives the series of interest in this problem.
     
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