1 + 2 + 4 + 8 + · · ·

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.

: $sum_{i=0}^{n} 2^i.$

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.

In the history and education of mathematics, nowrap|1 + 2 + 4 + 8 + … is the canonical example of a divergent geometric series with positive terms. Many results and arguments pertaining to the series have analogies with other examples such as nowrap|2 + 6 + 18 + 54 + ….

ummation

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 sum and Abel sum. [Hardy p.10]

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

: $f(x) = 1+2x+4x^2+8x^3+cdots+2^n{}x^n+cdots = frac{1}{1-2x}$

has a radius of convergence of only ¹/₂, 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" = ¹/₂ 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 approach to divergent series.) [Hardy pp.8, 10]

An almost identical approach is to consider the power series whose coefficients are all 1, i.e.

: $1+y+y^2+y^3+cdots = frac{1}{1-y}$

and plugging in "y" = 2. Of course these two series are related by the substitution "y" = 2"x".

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:

: $egin{array}{rcl}s & = &displaystyle 1+2+4+8+cdots \ [1em] & = &displaystyle 1+2(1+2+4+8+cdots) \ [1em] & = &displaystyle 1+2send{array}$

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 points 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. [The two roots of nowrap|1="s" = 1 + 2"s" are briefly touched on by Hardy p.19.]

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 integers appears to have the non-integer sum ¹⁄₂. 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 nowrap|1=0.111… = ¹⁄₉ and nowrap|1=0.999… = 1, but the underlying proofs demand careful thinking about the interpretation of endless sums. [Gardiner pp.93-99; the argument on p.95 for nowrap|1 + 2 + 4 + 8 + … is slightly different but has the same spirit.]

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. [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|id = ISBN 0-387-96017-1|year = 1984|pages = chapter I, exercise 16, p. 20]

Notes

See also

* Divergent geometric series
* 1 − 2 + 4 − 8 + · · ·

References

Further reading

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Academic Dictionaries and Encyclopedias

1 + 2 + 4 + 8 + · · ·

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

1 + 2 + 4 + 8 + · · ·

Share the article and excerpts

Direct link