1/2 + 1/4 + 1/8 + 1/16 + ⋯

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

{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =\sum _{n=1}^{\infty }\left({\frac {1}{2}}\right)^{n}=1.

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

Proof

As with any infinite series, the sum

{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots

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

s_{n}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots +{\frac {1}{2^{n-1}}}+{\frac {1}{2^{n}}}

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

s_{n}=1-{\frac {1}{2^{n}}}.

As $n$ approaches infinity, the term ${\frac {1}{2^{n}}}$ approaches 0 and so $s n$ tends to 1.

History

Zeno's paradox

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 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 the tortoise would already have moved another five meters by then. 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 Dichotomy paradox also states that to move a certain distance, you have to move half of it, then half of the remaining distance, and so 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 a finite number.

The Eye of Horus

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

In Zhuangzi

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]}

Notes

^ For example: multiplying $s n$ by 2 yields $2s_{n}={\frac {2}{2}}+{\frac {2}{4}}+{\frac {2}{8}}+{\frac {2}{16}}+\cdots +{\frac {2}{2^{n}}}=1+\left[{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{n-1}}}\right]=1+\left[s_{n}-{\frac {1}{2^{n}}}\right].$ Subtracting $s n$ from both sides, one concludes $s_{n}=1-{\frac {1}{2^{n}}}.$ Other arguments might proceed by mathematical induction, or by adding ${\frac {1}{2^{n}}}$ to both sides of $s_{n}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots +{\frac {1}{2^{n-1}}}+{\frac {1}{2^{n}}}$ and manipulating to show that the right side of the result is equal to 1.^{[citation needed]}

References

^ ^a ^b Field, Paul and Weisstein, Eric W. "Zeno's Paradoxes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ZenosParadoxes.html
^ Stewart, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. pp. 76–80. ISBN 978-1-84668-292-6.