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Math Help - Quickie #10

  1. #1
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    Quickie #10


    Prove that any positive rational number can be expressed as a finite sum

    . . of distinct terms of the harmonic series: . 1,\:\frac{1}{2},\:\frac{1}{3},\:\frac{1}{4},\:\hdo  ts,\:\frac{1}{n}

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  2. #2
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    I created my own Riddle.

    I was going to take the following approach....but then I stopped....why?

    We will show that \mathbb{Q} is a vector space over \mathbb{Z} spanned by the infinite dimensional set S=\{1,1/2,1/3,...\}. But then I stopped there. Why?

    ------------
    I am not going to answer your riddle because I already knew that, let someone else try.
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  3. #3
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    No one took up the challenge?

    This is an oldie from the study of "Eqyptian fractions".

    There is no hide/show feature at this site, so here's the solution.


    The rational number \frac{a}{b} can be written: . \underbrace{\frac{1}{b} + \frac{1}{b} + \frac{1}{b} + \cdots + \frac{1}{b}}_{a\text{ terms}}

    We eliminate the duplications by repeatedly using the identity:
    . . \frac{1}{n} \:=\:\frac{1}{n+1} + \frac{1}{n(n+1)} .until all denominators are distinct.


    Example: \frac{3}{7}

    \frac{3}{7} \;=\;\frac{1}{7}\quad\; +\quad \frac{1}{7}\qquad\quad\; +\qquad\quad\; \frac{1}{7}

    . . = \;\frac{1}{7} + \overbrace{\left(\frac{1}{8} + \frac{1}{56}\right)} +\quad \overbrace{\left(\frac{1}{8}\qquad +\qquad \frac{1}{56}\right)}

    . . = \;\frac{1}{7} + \left(\frac{1}{8} + \frac{1}{56}\right) + \overbrace{\left(\frac{1}{9} + \frac{1}{72}\right)} + \overbrace{\left(\frac{1}{57} + \frac{1}{3192}\right)}


    Therefore: . \frac{3}{7}\;=\;\frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{56} + \frac{1}{57} + \frac{1}{72} + \frac{1}{3192}

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  4. #4
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    Yes, it is called the Greedy algorithm.

    ---
    Anybody want to try my riddle above?
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