1/2 + 1/4 + 1/8 + 1/16 + ⋯
There are many different expressions that can be shown to be equivalent to the problem, such as the form: 2−1 + 2−2 + 2−3 + ...
The sum of this series can be denoted in summation notation as:
As with any infinite series, the infinite sum
is defined to mean the limit of the sum of the first n terms
as n approaches infinity.
Multiplying sn by 2 reveals a useful relationship:
Subtracting sn from both sides,
As n approaches infinity, sn tends to 1.
This series was used as a representation of many of Zeno's paradoxes, one of which, Achilles and the Tortoise, is shown here. In the paradox, The warrior Achilles was to race against a tortoise. Achilles could run at 10 m/s, while the tortoise only 5. The tortoise, with a 10 meter advantage, Zeno argued, would win. The 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.
The Eye of Horus
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