1 + 2 + 4 + 8 + ⋯ Explained

In mathematics, 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

1 + 2 + 4 + 8 + … are 1 , 3 , 7 , 15 , \ldots; since these diverge to infinity, so does the series. $$2^0+2^1\; +\; \backslash cdots\; +\; 2^k\; =\; 2^-1$$

It is written as

\sum ^{inf ty} _{n =0} 2 ^{n}

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

1 + 2 + 4 + 8 + … to the finite value of −1. The associated power series$$f(x)\; =\; 1\; +\; 2x\; +\; 4x^2\; +\; 8x^3+\; \backslash cdots\; +\; 2^nx^n\; +\; \backslash cdots\; =\; \backslash frac$$ has a radius of convergence around 0 of only (1)/(2) so it does not converge at x = 1 . Nonetheless, the so-defined function f has a unique analytic continuation to the complex plane with the point x = (1)/(2) deleted, and it is given by the same rule f(x) = (1)/(1-2 x) . Since f( 1 ) = -1 , the original series 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.)^[2]

An almost identical approach (the one taken by Euler himself) is to consider the power series whose coefficients are all 1, that is,$$1\; +\; y\; +\; y^2\; +\; y^3\; +\; \backslash cdots\; =\; \backslash frac$$and plugging in

y = 2 . These two series are related by the substitution y = 2 x.

The fact that (E) summation assigns a finite value to

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:\begin{array}{rcl} s & = &\displaystyle 1+2+4+8+16+ … \\ & = &\displaystyle 1+2 ( 1+2+4+8+ … ) \\ & = &\displaystyle 1+2 s \end{array}

In a useful sense,

s = inf ty is a root of the equation s = 1 + 2 s. (For example, inf ty is one of the two fixed points of the Möbius transformation z \mapsto 1 + 2 z on the Riemann sphere). If some summation method is known to return an ordinary number for s ; that is, not inf ty, then it is easily determined. In this case s may be subtracted from both sides of the equation, yielding 0 = 1 + s, so s = -1 . ^[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 + … (Grandi's series), 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 \ldots and most notably 0 . 999 \ldots . The arguments are ultimately justified for these convergent series, implying that 0 . 111 \ldots = (1)/(9) and 0 . 999 \ldots = 1 , 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 .

References

• Euler . Leonhard . Leonhard Euler . De seriebus divergentibus . Novi Commentarii Academiae Scientiarum Petropolitanae . 5 . 1760 . 205–237 .
• Book: Gardiner, A. . Anthony Gardiner (mathematician) . Understanding infinity: the mathematics of infinite processes . 2002 . 1982 . Dover . Dover . 0-486-42538-X.
• Book: Hardy, G. H. . G. H. Hardy . Divergent Series . 1949 . Clarendon Press . .

Further reading

• Barbeau . E. J. . P. J.. Leah . Euler's 1760 paper on divergent series . Historia Mathematica . 3 . 2 . 141–160 . 10.1016/0315-0860(76)90030-6. May 1976. free .
• Ferraro . Giovanni . Convergence and Formal Manipulation of Series from the Origins of Calculus to About 1730 . Annals of Science . 59 . 2002 . 2 . 179–199 . 10.1080/00033790010028179. 143992318 .
• Kline . Morris . Morris Kline . Euler and Infinite Series . Mathematics Magazine . 56 . 5 . 307–314 . 10.2307/2690371 . 2690371. November 1983.
• Web site: Sandifer . Ed . Divergent series . How Euler Did It . MAA Online . June 2006 . 2007-02-17 . 2013-03-20 . https://web.archive.org/web/20130320011704/http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2032%20divergent%20series.pdf . dead .
• Sierpińska . Anna . Anna Sierpińska. Humanities students and epistemological obstacles related to limits . Educational Studies in Mathematics . 18 . 4 . 371–396 . 10.1007/BF00240986 . 3482354. November 1987. 144880659 .

Notes and References

1. Hardy p. 10
2. Hardy pp. 8, 10
3. The two roots of s = 1 + 2 s are briefly touched on by Hardy p. 19.
4. Gardiner pp. 93–99; the argument on p. 95 for 1 + 2 + 4 + 8 + … is slightly different but has the same spirit.
5. Book: Koblitz, Neal. p-adic Numbers, p-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics, vol. 58. Springer-Verlag. 0-387-96017-1. 1984. chapter I, exercise 16, p. 20.