3 8 Plus 1 16

Mathematical infinite series

Beginning vi summands fatigued as portions of a square.

The geometric series on the existent line.

In mathematics, the space series one / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an uncomplicated example of a geometric serial that converges absolutely. The sum of the series is ane. In summation note, this may exist expressed as

${\displaystyle {\frac {1}{2}}+{\frac {1}{iv}}+{\frac {1}{8}}+{\frac {one}{16}}+\cdots =\sum _{northward=1}^{\infty }\left({\frac {1}{2}}\right)^{n}=ane.}$

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

Proof [edit]

As with whatsoever infinite serial, the sum

${\displaystyle {\frac {ane}{2}}+{\frac {1}{4}}+{\frac {1}{eight}}+{\frac {i}{16}}+\cdots }$

is defined to mean the limit of the fractional sum of the first northward terms

${\displaystyle s_{n}={\frac {1}{2}}+{\frac {1}{four}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots +{\frac {1}{ii^{n-i}}}+{\frac {one}{ii^{due north}}}}$

equally due north approaches infinity. Past diverse arguments,^[a] one can show that this finite sum is equal to

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

As due north approaches infinity, the term ${\displaystyle {\frac {one}{2^{north}}}}$ approaches 0 and so southward_n tends to 1.

History [edit]

Zeno's paradox [edit]

This series was used as a representation of many of Zeno's paradoxes.^[i] 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/southward, while the tortoise simply v. The tortoise, with a 10-meter advantage, Zeno argued, would win. Achilles would accept to move 10 meters to grab up to the tortoise, just the tortoise would already have moved some other five meters by then. Achilles would then take to move 5 meters, where the tortoise would move 2.5 meters, and then on. Zeno argued that the tortoise would always remain alee of Achilles.

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

The Middle of Horus [edit]

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

In a myriad ages it will non exist exhausted [edit]

A version of the series appears in the aboriginal Taoist volume Zhuangzi. The miscellaneous chapters "All Under Heaven" include the post-obit judgement: "Take a chi long stick and remove half every 24-hour interval, in a myriad ages it volition not be wearied."^{[ citation needed ]}

Run across also [edit]

• 0.999...
• 1/2 − 1/4 + i/8 − i/sixteen + ⋯
• Bodily infinity

Notes [edit]

References [edit]

1. ^ ^{a b} Field, Paul and Weisstein, Eric West. "Zeno'due south Paradoxes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ZenosParadoxes.html
2. ^ Stewart, Ian (2009). Professor Stewart'southward Hoard of Mathematical Treasures. Contour Books. pp. 76–80. ISBN978 1 84668 292 vi.

3 8 Plus 1 16,

Source: https://en.wikipedia.org/wiki/1/2_+_1/4_+_1/8_+_1/16_+_%E2%8B%AF

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