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

It is written as

\sum _{n=0}^{\infty }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 $1+2+4+8+\cdots$ 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 around 0 of only ${\frac {1}{2}}$ 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 $x={\frac {1}{2}}$ deleted, and it is given by the same rule $f(x)={\frac {1}{1-2x}}.$ Since $f(1)=-1,$ the original series $1+2+4+8+\cdots$ 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, $1+y+y^{2}+y^{3}+\cdots ={\frac {1}{1-y}}$ 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 $1+2+4+8+\cdots$ 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+\cdots \\&=&\displaystyle 1+2(1+2+4+8+\cdots )\\&=&\displaystyle 1+2s\end{array}}

In a useful sense, $s=\infty$ is a root of the equation $s=1+2s.$ (For example, $\infty$ is one of the two fixed points of the Möbius transformation $z\mapsto 1+2z$ on the Riemann sphere). If some summation method is known to return an ordinary number for $s$ ; that is, not $\infty ,$ 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 $1-1+1-1+\cdots$ (Grandi's series), where a series of integers appears to have the non-integer sum ${\frac {1}{2}}.$ 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 $0.111\ldots ={\frac {1}{9}}$ 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]

Notes

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

References

Euler, Leonhard (1760). "De seriebus divergentibus". Novi Commentarii Academiae Scientiarum Petropolitanae. 5: 205–237.
Gardiner, A. (2002) [1982]. Understanding infinity: the mathematics of infinite processes (Dover ed.). Dover. ISBN 0-486-42538-X.
Hardy, G. H. (1949). Divergent Series. Clarendon Press. LCC QA295 .H29 1967.

1 + 2 + 4 + 8 + ⋯

Contents

Summation

See also

Notes

References

Further reading