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Thread: Sigma Notation of Zeno's Pardadox

  1. #1
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    Sigma Notation of Zeno's Pardadox

    Achilles and the Tortoise Problem
    Tsub1 = 10
    Tsub2 = 1
    Tsub3 = .1
    Tsub4 = .01

    Sigma with
    infinity on top
    i = 1 on bottom
    tsubi on right

    i got that it = 10/(10^0) + 10/(10^1) + 10/(10^2).....

    so 0-10/9 so -(10/9)?
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  2. #2
    Super Member wingless's Avatar
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    $\displaystyle \sum^{\infty}_{k=1} 10^{2-k}$

    $\displaystyle \sum^{\infty}_{k=0} 10^{1-k}$

    $\displaystyle \sum^{\infty}_{k=0} \frac{10}{10^k}$

    $\displaystyle 10 \sum^{\infty}_{k=0} {\left( \frac{1}{10}\right)}^k$


    -----
    If $\displaystyle 0<r<1$, then $\displaystyle \sum^{\infty}_{k=0} r^k = \frac{1}{1-r}$
    -----

    So,
    $\displaystyle 10 \sum^{\infty}_{k=0} {\left( \frac{1}{10}\right)}^k = 10\cdot \frac{1}{1-\frac{1}{10}}$

    $\displaystyle 10\cdot\frac{1}{\frac{9}{10}} = \boxed{~\frac{100}{9}~}$
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  3. #3
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    how did you get what you started off with?
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  4. #4
    Super Member wingless's Avatar
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    Your sequence is
    $\displaystyle 10$
    $\displaystyle 1$
    $\displaystyle 0.1$
    $\displaystyle 0.01$
    ...

    You can write it as
    $\displaystyle 10^1$
    $\displaystyle 10^0$
    $\displaystyle 10^{-1}$
    $\displaystyle 10^{-2}$
    ...

    Do you see how to formulize the sequence now?
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