Archimedes 1.2.1

Archimedes and Continued Fractions. John G. Thompson University of Cambridge It is to Archimedes that we owe the inequalities The letter 1r is the first letter of the Greek word for perimeter, and is under­ stood to mean the circumference of the circle of diameter 1. On this page, the changelog for the mod Archimedes' Ships can be found. It can also be found on the mod Minecraft Forum thread.This page is almost a direct copy of the changelog on the Minecraft Forum thread, although it has been formatted and properly linked for the wiki.

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The Archimedes Cattle Problem In the third century BC, the famous Greek mathematician Archimedes issued a challenge to the Alexandrian mathematicians, headed by Eratosthenes. Written in the form of an epigram, Archimedes's challenge begins thus. This is '122EQUIREEL - Charlotte Faulkner & ARCHIMEDES X, TWESELDOWN (1)121' by Equireel on Vimeo, the home for high quality videos and the people. 4 days ago  Champions League: Bet £10 get £40 + Double the winnings with bet builder. Bet 10 get 40: New customers only. £10 deposit using promo code. Minimum stake £10 at odds of 1/2 (1.5). Free bets credited upon qualifying bet settlement and expire after 7 days. Free bet stakes not included in returns. Deposit balance is available for withdrawal at any time. Withdrawal restrictions. Archimedes lived during the Carboniferous period. It was a fenestrate bryozoan, that was much wider in life than it seems from the fossil. It was named for the Greek thinker Archimedes, who invented the water screw - Archimedes looks very much like a screw. It was a filter feeder, than was benthic and sessile in nature, living in shallow marine.

Archimedes 1.2.1 Free

(Redirected from 1/2 − 1/4 + 1/8 − 1/16 + · · ·)
Demonstration that 1/21/4 + 1/81/16 + ⋯ = 1/3

In mathematics, the infinite series1/21/4 + 1/81/16 + ⋯ is a simple example of an alternating series that converges absolutely.

It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is

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n=1(1)n+12n=1214+18116+=121(12)=13.{displaystyle sum _{n=1}^{infty }{frac {(-1)^{n+1}}{2^{n}}}={frac {1}{2}}-{frac {1}{4}}+{frac {1}{8}}-{frac {1}{16}}+cdots ={frac {frac {1}{2}}{1-(-{frac {1}{2}})}}={frac {1}{3}}.}

Hackenbush and the surreals[edit]

Archimedes 1.2.1 Video

A slight rearrangement of the series reads

11214+18116+=13.{displaystyle 1-{frac {1}{2}}-{frac {1}{4}}+{frac {1}{8}}-{frac {1}{16}}+cdots ={frac {1}{3}}.}

The series has the form of a positive integer plus a series containing every negative power of two with either a positive or negative sign, so it can be translated into the infinite blue-red Hackenbush string that represents the surreal number1/3:

LRRLRLR… = 1/3.[1]

A slightly simpler Hackenbush string eliminates the repeated R:

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LRLRLRL… = 2/3.[2]

In terms of the Hackenbush game structure, this equation means that the board depicted on the right has a value of 0; whichever player moves second has a winning strategy.

Related series[edit]

  • The statement that 1/21/4 + 1/81/16 + ⋯ is absolutely convergent means that the series 1/2 + 1/4 + 1/8 + 1/16 + ⋯ is convergent. In fact, the latter series converges to 1, and it proves that one of the binary expansions of 1 is 0.111….
  • Pairing up the terms of the series 1/21/4 + 1/81/16 + ⋯ results in another geometric series with the same sum, 1/4 + 1/16 + 1/64 + 1/256 + ⋯. This series is one of the first to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC.[3]
  • The Euler transform of the divergent series 1 − 2 + 4 − 8 + ⋯ is 1/21/4 + 1/81/16 + ⋯. Therefore, even though the former series does not have a sum in the usual sense, it is Euler summable to 1/3.[4]

Notes[edit]

  1. ^Berkelamp et al. p. 79
  2. ^Berkelamp et al. pp. 307–308
  3. ^Shawyer and Watson p. 3
  4. ^Korevaar p. 325

References[edit]

  • Berlekamp, E. R.; Conway, J. H.; Guy, R. K. (1982). Winning Ways for your Mathematical Plays. Academic Press. ISBN0-12-091101-9.
  • Korevaar, Jacob (2004). Tauberian Theory: A Century of Developments. Springer. ISBN3-540-21058-X.
  • Shawyer, Bruce; Watson, Bruce (1994). Borel's Methods of Summability: Theory and Applications. Oxford UP. ISBN0-19-853585-6.
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