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Summation Question (Properties)

  1. Dec 2, 2004 #1
    I have a rather simple question, but my rusty brain needs a good, swift kick-start.

    I start with:

    [tex]\sum_{i=1}^k i[/tex]

    and substitute in [tex]i=k-j[/tex] to get:

    [tex]\sum_{k-j=1}^k (k-j)[/tex]

    How do I get from this to the following?

    [tex]\sum_{k-j=1}^k (k-j) \rightarrow \sum_{j=0}^{k-1} (k-j)[/tex]

    Thanks in advance for your help.

    dogma
     
    Last edited: Dec 2, 2004
  2. jcsd
  3. Dec 2, 2004 #2

    arildno

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    Dearly Missed

    You're in some confusion in how to interpret the summation limits.
    Let's write it explicitly, to see how it follows:
    [tex]\sum_{i\geq{1}}^{i\leq{k}}i=\sum_{(k-j)\geq{1}}^{(k-j)\leq{k}}(k-j)[/tex]

    Now, rearrange the inequalities in the last expression:
    [tex]\sum_{(k-j)\geq{1}}^{(k-j)\leq{k}}(k-j)=\sum_{(k-1)\geq{j}}^{0\leq{j}}(k-j)[/tex]
    Which in standard notation is nothing else than:
    [tex]\sum_{j\geq{0}}^{j\leq(k-1)}(k-j)=\sum_{j=0}^{k-1}(k-j)[/tex]
     
    Last edited: Dec 2, 2004
  4. Dec 2, 2004 #3
    Thank you!

    I completely understand now. I just need a good, swift kick. :tongue2:

    Thanks again and take care!

    dogma
     
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