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Math Help - Summation division

  1. #1
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    Summation division

    Simple question. If you have a fraction where:

    numerator = summation of Xi terms
    denominator = summation of Xi^2 terms

    Do standard rules of arithmetic apply? So that it equals the summation of 1/Xi ?
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  2. #2
    Senior Member Paze's Avatar
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    Re: Summation division

    \frac{3i}{3i^2}\rightarrow 3i^-1\rightarrow \frac{3i}{3i^2}=\frac{1}{3i}

    Is this your question?

    Or are you asking something in the lines of: \frac{x+x^2+x^3}{x^2}=\frac{1}{x}+x+1

    ?

    P.S. I don't know how to make a negative exponent in latex. Only the - sign gets superscripted. Maybe someone can help me with that?
    Last edited by Paze; February 3rd 2013 at 10:59 AM.
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  3. #3
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    Re: Summation division

    Replace the 3 in the first term with the sigma notation, and that's my question i.e., what does that simplify to?
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  4. #4
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    Re: Summation division

    Quote Originally Posted by Paze View Post
    \frac{3i}{3i^2}\rightarrow 3i^{-1}\rightarrow \frac{3i}{3i^2}=\frac{1}{3i}
    P.S. I don't know how to make a negative exponent in latex. Only the - sign gets superscripted. Maybe someone can help me with that?
    [TEX]3i^{-1}[/TEX] gives 3i^{-1}.

    If an exponent has more that one character in it use {}.
    It is a good habit to use {} for all exponents.
    Thanks from Paze
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  5. #5
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    Re: Summation division

    Paze, to get both 3i^{-1}, put the entire exponent, -1, in braces: {-1}.

    MN1987, Paze is looking at the case of one term and I think he means \frac{3i}{(3i)^2}= \frac{1}{3i}

    However, you seem to be talking about \frac{\sum x_i}{\sum (x_i)^2} which is completely differenct. For example, if x_i= i, with i from 1 to 5, that becomes \frac{1+ 2+ 3+ 4+ 5}{1+ 4+ 9+ 16+ 25}= \frac{15}{55}= \frac{3}{11} while if x_1, x_2= 4, x_3= 5, x_4= 9, x_5= 12 (chosen pretty much at random) then \frac{\sum x_i}{\sum x_i^2}= \frac{1+ 4+ 5+ 9+ 12}{1+ 16+ 25+ 81+ 144}= \frac{31}{167}.
    There simply is no simple formula for a ratio of two sums. Addition and multiplicatio do not "play well together"!
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  6. #6
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    Re: Summation division

    Quote Originally Posted by MN1987 View Post
    Simple question. If you have a fraction where:
    numerator = summation of Xi terms
    denominator = summation of Xi^2 terms
    Do standard rules of arithmetic apply? So that it equals the summation of 1/Xi ?
    We assume that you mean finite sums.

    \frac{{\sum\limits_{k = 1}^n {x_k } }}{{\sum\limits_{k = 1}^n {\left( {x_k } \right)^2 } }}

    Let S = \sum\limits_{k = 1}^n {x_k } \;\& \,T = \sum\limits_{k = 1}^n {\left( {x_k } \right)^2 }

    then \[\frac{{\sum\limits_{k = 1}^n {x_k } }}{{\sum\limits_{k = 1}^n {\left( {x_k } \right)^2 } }} = \frac{{\sum\limits_{k = 1}^n {x_k } }}{T} = \sum\limits_{k = 1}^n {\frac{{x_k }}{T}}

    Is that what you mean?
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  7. #7
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    Re: Summation division

    Wow, I had no idea I had phrased my original question to be so ambiguous! I need to learn to use Latex notation.

    Yes, I meant:
    \frac{{\sum\limits_{i = 1}^n {x_i } }}{{\sum\limits_{i = 1}^n {\left( {x_i } \right)^2 } }}

    I guess the consensus is that it does not simplify further than that, for all intents and purposes. It's obvious now looking at it... shame! It would've made my assignment problem alot easier to solve
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