1 + 2 + 4 + 8 + ⋯
In mathematics, Template: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 in the usual sense it has no sum. In a much broader sense, the series is associated with another value besides ∞, namely −1, which is the limit of the series using the 2-adic metric.
Summation
The partial sums of Template:Nowrap are Template:Nowrap since these diverge to infinity, so does the series.
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 Template:Nowrap to the finite value of −1. The associated power series
has a radius of convergence around 0 of only Template:Sfrac, so it does not converge at Template:Nowrap. Nonetheless, the so-defined function f has a unique analytic continuation to the complex plane with the point Template:Nowrap deleted, and it is given by the same rule Template:Nowrap. Since Template:Nowrap, the original series Template:Nowrap 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, i.e.
and plugging in y = 2. These two series are related by the substitution y = 2x.
The fact that (E) summation assigns a finite value to Template:Nowrap 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:
In a useful sense, s = ∞ is a root of the equation Template:Nowrap (For example, ∞ is one of the two fixed points of the Möbius transformation Template:Nowrap 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 Template:Nowrap, so Template:Nowrap.[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 1 − 1 + 1 − 1 + ⋯, where a series of integers appears to have the non-integer sum Template:Sfrac. These examples illustrate the potential danger in applying similar arguments to the series implied by such recurring decimals as 0.111… and most notably 0.999…. The arguments are ultimately justified for these convergent series, implying that Template:Nowrap and Template:Nowrap, 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 + · · ·
- 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 Template:Nowrap.
Notes
References
Further reading
- Template:Cite journal
- Template:Cite journal
- Template:Cite journal
- Template:Cite web
- Template:Cite journal
- ↑ Hardy p. 10
- ↑ Hardy pp.8, 10
- ↑ The two roots of Template:Nowrap are briefly touched on by Hardy p. 19.
- ↑ Gardiner pp. 93–99; the argument on p. 95 for Template:Nowrap is slightly different but has the same spirit.
- ↑ Template:Cite book