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I recall from undergraduate courses in calculus and series analysis a tale of a frog that tries to jump a fraction (e.g. 1/2) of what is left for the frog to cross the pond.

In the limit, the fraction of the pond the frog travels is:

$1/2 + 1/2(1/2) + 1/2 (1/4) + ...$

Does this tale have a name? What about the series?

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    $\begingroup$ Generally, it's a "geometric series". I don't know of a specific name for this one though. $\endgroup$ Commented Aug 7, 2015 at 13:43
  • $\begingroup$ Note: The sum of the series is $a(1-r^n)/(1-r)$ or for the infinite case, $a/(1-r)$. $\endgroup$
    – Kartik
    Commented Aug 7, 2015 at 13:51
  • $\begingroup$ @Kartik In this infinite case this is true only if $|r| < 1$. Otherwise the series diverges. $\endgroup$
    – eigenchris
    Commented Aug 7, 2015 at 13:55
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    $\begingroup$ Achilles and the tortoise, one of Zeno's paradoxes : en.wikipedia.org/wiki/Zeno%27s_paradoxes $\endgroup$ Commented Aug 7, 2015 at 13:56
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    $\begingroup$ There’s also the variant in which the boys line up on one side of the room, the girls on the other, and the boys advance half the remaining distance each minute; they never reach the girls, but ‘they get close enough for all practical purposes’. $\endgroup$ Commented Aug 7, 2015 at 14:08

2 Answers 2

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This looks like a geometric series, which is a series of the form

$$\sum_{n=0}^\infty a r^n$$

In your case $a = \frac{1}{2}$ and $r = \frac{1}{2}$.

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As one of the contents mentions, this story shares most of its properties with Zeno's paradoxes. It seems similar in its description to Douglas Hofstadter's explanation of Zeno's paradoxes in his book Gödel Escher Bach.

As the other answer says, mathematically it's a geometric series.

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