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Infinite Series

  1. Aug 27, 2010 #1
    I know that

    [tex]
    \sum_{n=-\infty}^\infty{1} = \infty
    [/tex]

    But I don't understand why.

    It seems to me that since the constant inside the summation is not dependent upon n it can be moved outside the summation. Then there is nothing to sum.


    It seems to me that

    [tex]
    \sum_{n=-\infty}^\infty{1}
    [/tex]

    should equal 1.

    What am I missing?
     
  2. jcsd
  3. Aug 27, 2010 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    Your logic is incorrect. The sum is 1+1+1+1+.... no matter how you slice it.

    You can take the 1 outside the sum, but you still have 1 inside, not 0. 1x0=0, not =1.
     
  4. Aug 27, 2010 #3
    I thought the summation just vanished if there was no "argument" inside it.

    That makes a lot more sense of why the summation is a discrete analogue of an integral.

    Thank you for the help.
     
  5. Aug 27, 2010 #4
  6. Aug 27, 2010 #5

    CRGreathouse

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    Science Advisor
    Homework Helper

    You can change
    [tex]\sum_{n=1}^N1[/tex]
    to
    [tex](N\cdot1)+\sum_{n=1}^N0=N[/tex]
    by 'pulling out the 1'. In your infinite sum, proceeding formally, this would give you
    [tex]\sum_{n=1}^\infty1[/tex]
    to
    [tex](\infty\cdot1)+\sum_{n=1}^\infty0=\infty[/tex]
    which shows (in a non-rigorous way) that the sum diverges.
     
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