# 1 + 2 + 4 + 8 + ⋯

The first four partial sums of 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 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 ${\displaystyle 1+2+4+8+\cdots }$ are ${\displaystyle 1,3,7,15,\ldots ;}$ since these diverge to infinity, so does the series.

${\displaystyle 2^{0}+2^{1}+\cdots +2^{k}=2^{k+1}-1}$

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 ${\displaystyle 1+2+4+8+\cdots }$ to the finite value of −1. The associated power series

${\displaystyle f(x)=1+2x+4x^{2}+8x^{3}+\cdots +2^{n}{}x^{n}+\cdots ={\frac {1}{1-2x}}}$
has a radius of convergence around 0 of only ${\displaystyle {\frac {1}{2}}}$ so it does not converge at ${\displaystyle x=1.}$ Nonetheless, the so-defined function ${\displaystyle f}$ has a unique analytic continuation to the complex plane with the point ${\displaystyle x={\frac {1}{2}}}$ deleted, and it is given by the same rule ${\displaystyle f(x)={\frac {1}{1-2x}}.}$ Since ${\displaystyle f(1)=-1,}$ the original series ${\displaystyle 1+2+4+8+\cdots }$ 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,

${\displaystyle 1+y+y^{2}+y^{3}+\cdots ={\frac {1}{1-y}}}$
and plugging in ${\displaystyle y=2.}$ These two series are related by the substitution ${\displaystyle y=2x.}$

The fact that (E) summation assigns a finite value to ${\displaystyle 1+2+4+8+\cdots }$ 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:

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

In a useful sense, ${\displaystyle s=\infty }$ is a root of the equation ${\displaystyle s=1+2s.}$ (For example, ${\displaystyle \infty }$ is one of the two fixed points of the Möbius transformation ${\displaystyle z\mapsto 1+2z}$ on the Riemann sphere). If some summation method is known to return an ordinary number for ${\displaystyle s}$; that is, not ${\displaystyle \infty ,}$ then it is easily determined. In this case ${\displaystyle s}$ may be subtracted from both sides of the equation, yielding ${\displaystyle 0=1+s,}$ so ${\displaystyle s=-1.}$[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 ${\displaystyle 1-1+1-1+\cdots }$ (Grandi's series), where a series of integers appears to have the non-integer sum ${\displaystyle {\frac {1}{2}}.}$ These examples illustrate the potential danger in applying similar arguments to the series implied by such recurring decimals as ${\displaystyle 0.111\ldots }$ and most notably ${\displaystyle 0.999\ldots }$. The arguments are ultimately justified for these convergent series, implying that ${\displaystyle 0.111\ldots ={\frac {1}{9}}}$ and ${\displaystyle 0.999\ldots =1,}$ 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]

3. ^ The two roots of ${\displaystyle s=1+2s}$ are briefly touched on by Hardy p. 19.
4. ^ Gardiner pp. 93–99; the argument on p. 95 for ${\displaystyle 1+2+4+8+\cdots }$ is slightly different but has the same spirit.