How to combine a summation

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Hint: Note that every $f(a,b)$ with $a\gt 0$ and $b \gt 0$ is part of both summations. But $f(0,b)$ and $f(a,0)$ are only represented once, and $f(0,0)$ doesn't contribute at all. This occurs often with summations-you need to think about getting every term once and only once in your rewrite.

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user1947180
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Updated on June 16, 2020

Comments

  • user1947180
    user1947180 over 3 years

    Is there a way to combine this into one summation or is it not possible:

    $$\sum_{a=0}^A\sum_{b=1}^B f(a,b) + \sum_{a=1}^A\sum_{b=0}^B f(a,b)$$

  • user1947180
    user1947180 over 10 years
    I tried that (combining everything twice and then subtracting out the 0,0 case) but nothing I try is working
  • Ross Millikan
    Ross Millikan over 10 years
    @user1947180: I said some terms come in only once. Maybe if you show your work we can find the problem.
  • user1947180
    user1947180 over 10 years
    Is the only way to simplify it the sum from a=1 and b=1 and then add on the sums assuming a=0 and then when b=0?
  • Ross Millikan
    Ross Millikan over 10 years
    @user1947180: only is a difficult word. That approach works. In some cases, there might be another. You have found the inclusion-exclusion principle, which is an extremely useful technique.
  • user1947180
    user1947180 over 10 years
    Is there a "cleaner" way to simplify the expression rather than expressing it as a three-part sum?
  • Ross Millikan
    Ross Millikan over 10 years
    @user1947180: Not that I know in general. There may be features of $f(a,b)$ that allow it.