# Tale of a frog that jumps a fraction of what is left to cross the pond

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?

• Generally, it's a "geometric series". I don't know of a specific name for this one though. Commented Aug 7, 2015 at 13:43
• Note: The sum of the series is $a(1-r^n)/(1-r)$ or for the infinite case, $a/(1-r)$. Commented Aug 7, 2015 at 13:51
• @Kartik In this infinite case this is true only if $|r| < 1$. Otherwise the series diverges. Commented Aug 7, 2015 at 13:55
• Achilles and the tortoise, one of Zeno's paradoxes : en.wikipedia.org/wiki/Zeno%27s_paradoxes Commented Aug 7, 2015 at 13:56
• 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’. Commented Aug 7, 2015 at 14:08

$$\sum_{n=0}^\infty a r^n$$
In your case $a = \frac{1}{2}$ and $r = \frac{1}{2}$.