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| In [[mathematics]], '''1 + 2 + 4 + 8 + …''' is the [[infinite series]] whose terms are the successive [[powers of two]]. As a [[geometric series]], it is characterized by its first term, 1, and its common ratio, 2. As a series of [[real number]]s it [[divergent series|diverges]] to [[infinity]], so in the usual sense it has no sum. In a much broader sense, the series is associated with another value besides ∞, namely −1.
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| ==Summation==
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| The partial sums of 1 + 2 + 4 + 8 + … are {{nowrap|1, 3, 7, 15, …;}} since these diverge to infinity, so does the series. Therefore any [[totally regular summation method]] gives a sum of infinity, including the [[Cesàro summation|Cesàro sum]] and [[Abel summation|Abel sum]].<ref>Hardy p.10</ref> On the other hand, there is at least one generally useful method that sums {{nowrap|1 + 2 + 4 + 8 + …}} to the finite value of −1. The associated [[power series]]
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| :<math>f(x) = 1+2x+4x^2+8x^3+\cdots+2^n{}x^n+\cdots = \frac{1}{1-2x}</math>
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| has a [[radius of convergence]] around 0 of only <sup>1</sup>/<sub>2</sub>, so it does not converge at {{nowrap|1=''x'' = 1}}. Nonetheless, the so-defined function ''f'' has a unique [[analytic continuation]] to the [[complex plane]] with the point {{nowrap|1=''x'' = <sup>1</sup>/<sub>2</sub>}} deleted, and it is given by the same rule {{nowrap|1=''f''(x) = 1/(1 − 2''x'')}}. Since {{nowrap|1=''f''(1) = −1}}, the original series {{nowrap|1 + 2 + 4 + 8 + …}} is said to be summable (''E'') to −1, and −1 is the (''E'') sum of the series. (The notation is due to [[G. H. Hardy]] in reference to [[Leonhard Euler]]'s [[Euler on infinite series|approach to divergent series]].)<ref>Hardy pp.8, 10</ref>
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| An almost identical approach (the one taken by Euler himself) is to consider the power series whose coefficients are all 1, i.e.
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| :<math>1+y+y^2+y^3+\cdots = \frac{1}{1-y}</math>
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| and plugging in ''y'' = 2. Of course these two series are related by the substitution ''y'' = 2''x''.
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| The fact that (''E'') summation assigns a finite value to {{nowrap|1 + 2 + 4 + 8 + …}} shows that the general method is not totally regular. On the other hand, it possesses some other desirable qualities for a summation method, including stability and linearity. These latter two axioms actually force the sum to be −1, since they make the following manipulation valid:
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| :<math>\begin{array}{rcl}
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| s & = &\displaystyle 1+2+4+8+\cdots \\[1em]
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| & = &\displaystyle 1+2(1+2+4+8+\cdots) \\[1em]
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| & = &\displaystyle 1+2s
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| \end{array}</math>
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| In a useful sense, ''s'' = ∞ is a root of the equation {{nowrap|1=''s'' = 1 + 2''s''.}} (For example, ∞ is one of the two [[Fixed point (mathematics)|fixed point]]s of the [[Möbius transformation]] {{nowrap|1=''z'' → 1 + 2''z''}} on the [[Riemann sphere]].) If some summation method is known to return an ordinary number for ''s'', ''i.e.'' not ∞, then it is easily determined. In this case ''s'' may be subtracted from both sides of the equation, yielding {{nowrap|1=0 = 1 + ''s''}}, so {{nowrap|1=''s'' = −1}}.<ref>The two roots of {{nowrap|1=''s'' = 1 + 2''s''}} are briefly touched on by Hardy p.19.</ref>
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| The above manipulation might be called on to produce −1 outside of the context of a sufficiently powerful summation procedure. For the most well-known and straightforward sum concepts, including the fundamental convergent one, it is absurd that a series of positive terms could have a negative value. A similar phenomenon occurs with the divergent geometric series [[1 − 1 + 1 − 1 + · · ·]], where a series of [[integer]]s appears to have the non-integer sum <sup>1</sup>⁄<sub>2</sub>. These examples illustrate the potential danger in applying similar arguments to the series implied by such [[recurring decimal]]s as 0.111… and most notably [[0.999...|0.999…]]. The arguments are ultimately justified for these convergent series, implying that {{nowrap|1=0.111… = <sup>1</sup>⁄<sub>9</sub>}} and {{nowrap|1=0.999… = 1}}, but the underlying [[mathematical proof|proof]]s demand careful thinking about the interpretation of endless sums.<ref>Gardiner pp. 93–99; the argument on p.95 for {{nowrap|1 + 2 + 4 + 8 + …}} is slightly different but has the same spirit.</ref>
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| It is also possible to view this series as convergent in a number system different from the real numbers, namely, the [[P-adic number|2-adic numbers]]. As a series of 2-adic numbers this series converges to the same sum, −1, as was derived above by analytic continuation.<ref>{{cite book|author = Koblitz, Neal|title = p-adic Numbers, p-adic Analysis, and Zeta-Functions|series = Graduate Texts in Mathematics, vol. 58|publisher = Springer-Verlag|isbn = 0-387-96017-1|year = 1984|pages = chapter I, exercise 16, p. 20}}</ref>
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| == See also ==
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| * [[1 − 1 + 2 − 6 + 24 − 120 + · · ·]]
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| * [[1 − 2 + 3 − 4 + · · ·]]
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| * [[Two's complement]], a data convention for representing negative numbers where <math>-1</math> is represented as if it were <math>1+2+4+\ldots+2^{n-1}</math>.
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| == Notes ==
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| {{reflist}}
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| ==References==
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| {{refbegin}}
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| *{{cite journal |last=Euler |first=Leonhard |authorlink=Leonhard Euler |title=De seriebus divergentibus |journal=Novi Commentarii academiae scientiarum Petropolitanae |volume=5 |year=1760 |pages=205–237 |url=http://www.math.dartmouth.edu/~euler/pages/E247.html}}
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| *{{cite book |last=Gardiner |first=A. |authorlink=Anthony Gardiner (mathematician) |title=Understanding infinity: the mathematics of infinite processes |year=2002 |origyear=1982 |edition=Dover |publisher=Dover |isbn=0-486-42538-X}}
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| *{{cite book |last=Hardy |first=G.H. |authorlink=G. H. Hardy |title=Divergent Series |year=1949 |publisher=Clarendon Press |id={{LCC|QA295|.H29|1967}}}}
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| {{refend}}
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| ==Further reading==
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| {{refbegin}}
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| *{{cite journal |author=Barbeau, E.J., and P.J. Leah |title=Euler's 1760 paper on divergent series |journal=Historia Mathematica |volume=3 |issue=2 |pages=141–160 |doi=10.1016/0315-0860(76)90030-6|date=May 1976}}
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| *{{cite journal |last=Ferraro |first=Giovanni |title=Convergence and Formal Manipulation of Series from the Origins of Calculus to About 1730 |journal=Annals of Science |volume=59 |year=2002 |pages=179–199 |doi=10.1080/00033790010028179}}
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| *{{cite journal |last=Kline |first=Morris |authorlink=Morris Kline |title=Euler and Infinite Series |journal=Mathematics Magazine |volume=56 |issue=5 |pages=307–314 |doi=10.2307/2690371 |jstor=2690371|date=November 1983}}
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| *{{cite web |last=Sandifer |first=Ed |title=Divergent series |work=How Euler Did It |publisher=MAA Online |url=http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2032%20divergent%20series.pdf|format=PDF|date=June 2006}}
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| *{{cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |pages=371–396 |doi=10.1007/BF00240986 |jstor=3482354|date=November 1987}}
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| {{refend}}
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| {{Series (mathematics)}}
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| {{DEFAULTSORT:1 + 2 + 4 + 8 + ...}}
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| [[Category:Binary arithmetic]]
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| [[Category:Divergent series]]
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| [[Category:Geometric series]]
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| [[zh:1+2+4+8+…]]
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