Английская Википедия:1 + 2 + 4 + 8 + ⋯
In mathematics, Шаблон:Nowrap 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 numbers it diverges to infinity, so the sum of this series is infinity.
However, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. For example, the Ramanujan summation of this series is −1, which is the limit of the series using the 2-adic metric.
Summation
The partial sums of <math>1 + 2 + 4 + 8 + \cdots</math> are <math>1, 3, 7, 15, \ldots;</math> since these diverge to infinity, so does the series. <math display="block">2^0+2^1 + \cdots + 2^k = 2^{k+1}-1</math>
It is written as :<math> \sum_{n=0}^\infty 2^n </math>
Therefore, any totally regular summation method gives a sum of infinity, including the Cesàro sum and Abel sum.[1] On the other hand, there is at least one generally useful method that sums <math>1 + 2 + 4 + 8 + \cdots</math> to the finite value of −1. The associated power series <math display="block">f(x) = 1 + 2x + 4x^2 + 8x^3+ \cdots + 2^n{}x^n + \cdots = \frac{1}{1-2x}</math> has a radius of convergence around 0 of only <math>\frac{1}{2}</math> so it does not converge at <math>x = 1.</math> Nonetheless, the so-defined function <math>f</math> has a unique analytic continuation to the complex plane with the point <math>x = \frac{1}{2}</math> deleted, and it is given by the same rule <math>f(x) = \frac{1}{1 - 2 x}.</math> Since <math>f(1) = -1,</math> the original series <math>1 + 2 + 4 + 8 + \cdots</math> 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 approach to divergent series).[2]
An almost identical approach (the one taken by Euler himself) is to consider the power series whose coefficients are all 1, that is, <math display="block">1 + y + y^2 + y^3 + \cdots = \frac{1}{1-y}</math> and plugging in <math>y = 2.</math> These two series are related by the substitution <math>y = 2 x.</math>
The fact that (E) summation assigns a finite value to <math>1 + 2 + 4 + 8 + \cdots</math> 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:
- <math>\begin{array}{rcl}
s & = &\displaystyle 1+2+4+8+16+\cdots \\
& = &\displaystyle 1+2(1+2+4+8+\cdots) \\ & = &\displaystyle 1+2s
\end{array}</math>
In a useful sense, <math>s = \infty</math> is a root of the equation <math>s = 1 + 2 s.</math> (For example, <math>\infty</math> is one of the two fixed points of the Möbius transformation <math>z \mapsto 1 + 2 z</math> on the Riemann sphere). If some summation method is known to return an ordinary number for <math>s</math>; that is, not <math>\infty,</math> then it is easily determined. In this case <math>s</math> may be subtracted from both sides of the equation, yielding <math>0 = 1 + s,</math> so <math>s = -1.</math>[3]
The above manipulation might be called on to produce −1 outside 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 <math>1 - 1 + 1 - 1 + \cdots</math> (Grandi's series), where a series of integers appears to have the non-integer sum <math>\frac{1}{2}.</math> These examples illustrate the potential danger in applying similar arguments to the series implied by such recurring decimals as <math>0.111\ldots</math> and most notably <math>0.999\ldots</math>. The arguments are ultimately justified for these convergent series, implying that <math>0.111\ldots = \frac{1}{9}</math> and <math>0.999\ldots = 1,</math> but the underlying proofs demand careful thinking about the interpretation of endless sums.[4]
It is also possible to view this series as convergent in a number system different from the real numbers, namely, the 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.[5]
See also
- 1 − 1 + 2 − 6 + 24 − 120 + ⋯
- 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
- 1 + 1 + 1 + 1 + ⋯
- 1 − 2 + 3 − 4 + ⋯
- 1 + 2 + 3 + 4 + ⋯
- 1 − 2 + 4 − 8 + ⋯
- Two's complement, a data convention for representing negative numbers where −1 is represented as if it were Шаблон:Nowrap.
Notes
References
Further reading
