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Why this expression is true

  1. May 3, 2009 #1
    [tex]
    \sum_{n=1}^{\infty}\frac{\ln n}{n}> \sum_{n=1}^{\infty}\frac{1}{n}
    [/tex]

    ln is not always bigger then 1
    so when i am doing the comparing test
    i cant use that
    because ln 1 =0

    ??
     
    Last edited: May 3, 2009
  2. jcsd
  3. May 3, 2009 #2

    Mark44

    Staff: Mentor

    ln n > 1 for almost every positive number n. Compare the graphs of y = ln x and y = 1 and you'll see that what I'm saying is true.
     
  4. May 4, 2009 #3
    you said yourself "almost" not absolutely
     
  5. May 4, 2009 #4

    diazona

    User Avatar
    Homework Helper

    Remember that that sum is really just a whole list of terms, all added up. So you can split the sum into two sums, or write out some of the terms explicitly if you want. Just to give an example:

    [tex]\sum_{n=0}^{\infty} n^2 e^{-n} = \left(\sum_{n=0}^{6} n^2 e^{-n}\right) + \left(\sum_{n=7}^{\infty} n^2 e^{-n}\right) = 0^2 e^{-0} + 1^2 e^{-1} + 2^2 e^{-2} + \sum_{n=3}^{\infty} n^2 e^{-n}[/tex]

    You can do something like this for one or both of the sums in your expression, and it should help.
     
  6. May 4, 2009 #5

    Mark44

    Staff: Mentor

    In an infinite series, you can always ignore a finite number of terms without affecting whether the series converges or diverges.
     
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