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Rewriting a sum

  1. Nov 8, 2008 #1
    Hi all,

    I have the following sum

    [tex]\sum n - n_1 + 1 [/tex]

    which I split up in two independent sums

    [tex]\sum_{n_1=0}^N n + 1 - \sum_{n_1=0}^N n_1[/tex]

    the last sum can be written as

    [tex]0.5*n(n+1) [/tex]

    but how to rewrite the first sum any hints appreciated. The final answer is
    0.5(n+1)(n+2)

    but as stated above I have some problems getting there.
    Any help or advice appreciated thanks in advance
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 8, 2008 #2

    CompuChip

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    Homework Helper

    Unless you are holding something back,
    [tex] \sum_{n_1=0}^N n + 1 = (N + 1)(n + 1)[/tex]

    Further,
    [tex]\sum_{n_1=0}^N n_1 = \frac12 N (N + 1)[/tex]
    (note the capital N); assuming that in the first line,
    [tex] \sum n - n_1 + 1 [/tex]
    you meant
    [tex]\sum_{n_1 = 1}^N n - n_1 + 1 [/tex]
    you are otherwise more or less correct...

    Your notation is confusing though.
     
  4. Nov 8, 2008 #3
    thanks for the reply - my problem is understanding the first summation

    [tex](N+1)(n+1)[/tex]

    how is small n to be interpreted?

    Thanks in advance any hints appreciated.
     
  5. Nov 8, 2008 #4

    HallsofIvy

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    However YOU mean it! You wrote
    [tex]\sum_{n1= 0}^N (n+1)[/tex]
    The "index" is n1 and that changes from 0 to N, but there is no "n1" in the sum itself- you are just adding the number n+ 1 to itself N+1 times. Any number added to itself N+1 times is just N+1 times that number: here (N+1)(n+1).

    Actually it seems peculiar to me to use "n1" as an index. Why the 1? You are, of course, welcome to use whatever labels you like but I would have thought that [itex]\sum n- n1+ 1[/itex] would be interpreted as
    [tex]\sum_{n=0}^N n- n1+ 1[/itex]
    where n1 is some fixed number.
     
  6. Nov 8, 2008 #5
    thanks,

    n_1 can change its value that is why I write it like that. So in order to get to

    0.5(n+1)(n+2)

    I have to say that in the limit n = N?

    thanks in advance
     
  7. Nov 8, 2008 #6

    HallsofIvy

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    What limit are you talking about? Perhaps it would be a good idea if you stated exactly what the problem you are working on is! You have already been given the answer to the problem you stated.
     
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