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.

First six summands drawn as portions of a square.

The geometric series on the real line.

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:

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

Proof

As with any infinite series, the infinite sum

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

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

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

as n approaches infinity.

Multiplying s_n by 2 reveals a useful relationship:

${\displaystyle 2s_{n}={\frac {2}{2}}+{\frac {2}{4}}+{\frac {2}{8}}+{\frac {2}{16}}+\cdots +{\frac {2}{2^{n}}}=1+\left[{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{n-1}}}\right]=1+\left[s_{n}-{\frac {1}{2^{n}}}\right].}$

Subtracting s_n from both sides,

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

As n approaches infinity, s_n tends to 1.

History

Zeno's paradox

This series was used as a representation of many of Zeno's paradoxes, one of which, Achilles and the Tortoise, is shown here.[1] In the paradox, 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.

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 a myriad ages it will not be exhausted

"Zhuangzi", also known as "South China Classic", written by Zhuang Zhou. In the miscellaneous chapters ""All Under Heaven"", he said: "Take a chi long stick and remove half every day, in a myriad ages it will not be exhausted."

See also

• 0.999...
• Dotted note
• Calculus
• Power series
• Actual infinity

References

1. Wachsmuth, Bet G. "Description of Zeno's paradoxes". Archived from the original on 2014-12-31. Retrieved 2014-12-29.
2. Stewart, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. pp. 76–80. ISBN 978 1 84668 292 6.