This series was used as a representation of many of Zeno's paradoxes. For example, in the paradox of Achilles and the Tortoise, the warrior Achilles was to race against a tortoise. The track is 100 meters long. Achilles could run at 10 m/s, while the tortoise only 5. The tortoise, with a 10-meter advantage, Zeno argued, would win. Achilles would have to move 10 meters to catch up to the tortoise, but by then, the tortoise would already have moved another five meters. Achilles would then have to move 5 meters, where the tortoise would move 2.5 meters, and so on. Zeno argued that the tortoise would always remain ahead of Achilles.
A version of the series appears in the ancient Taoist book Zhuangzi. The miscellaneous chapters "All Under Heaven" include the following sentence: "Take a chi long stick and remove half every day, in a myriad ages it will not be exhausted."
^Stewart, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. pp. 76–80. ISBN978 1 84668 292 6.
^For example: multiplying sn by 2 yields Subtracting sn from both sides, one concludes Other arguments might proceed by mathematical induction, or by adding to both sides of and manipulating to show that the right side of the result is equal to 1.