1/2 + 1/4 + 1/8 + 1/16 + ⋯

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In mathematics, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as

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The series is related to lớn philosophical questions considered in antiquity, particularly to lớn Zeno's paradoxes.

Proof[edit]

As with any infinite series, the sum

is defined to lớn mean the limit of the partial sum of the first n terms

as n approaches infinity. By various arguments,^[a] one can show that this finite sum is equal to

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As n approaches infinity, the term approaches 0 and ví s_n tends to lớn 1.

History[edit]

Zeno's paradox[edit]

This series was used as a representation of many of Zeno's paradoxes.^[1] For example, in the paradox of Achilles and the Tortoise, the warrior Achilles was to lớn race against a tortoise. The track is 100 meters long. Achilles could run rẩy at 10 m/s, while the tortoise only 5. The tortoise, with a 10-meter advantage, Zeno argued, would win. Achilles would have to lớn move 10 meters to lớn catch up to lớn the tortoise, but the tortoise would already have moved another five meters by then. Achilles would then have to lớn move 5 meters, where the tortoise would move 2.5 meters, and ví on. Zeno argued that the tortoise would always remain ahead of Achilles.

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The Dichotomy paradox also states that to lớn move a certain distance, you have to lớn move half of it, then half of the remaining distance, and ví on, therefore having infinitely many time intervals.^[1] This can be easily resolved by noting that each time interval is a term of the infinite geometric series, and will sum to lớn a finite number.

The Eye of Horus[edit]

The parts of the Eye of Horus were once thought to lớn represent the first six summands of the series.^[2]

In a myriad ages it will not be exhausted[edit]

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."^{[citation needed]}