1 + 2 + 4 + 8 + ⋯

Summation

The partial sum of the first n terms of 1 + 2 + 4 + 8 + \cdots is \sum_{k=0}^{n - 1} 2^k = 2^0+2^1 + \cdots + 2^{n - 1} = 2^n - 1. Since the sequence 1, 3, 7, 15, \ldots of these partial sums diverges to infinity, so does the series. Therefore, any totally regular summation method gives a sum of infinity, including the Cesàro sum and Abel sum. 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 - 2 x}. 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.)

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 = 2 x.

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{align} s = {} & 1+2+4+8+16+\cdots \ = {} & 1+2(1+2+4+8+\cdots) \ = {} & 1+2s \end{align}

In a useful sense, s = \infty is a root of the equation s = 1 + 2 s. (For example, \infty 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 \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.

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.

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.

Notes

References

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 $1\; +\; 2\; +\; 4\; +\; 8\; +\; \backslash cdots$ is slightly different but has the same spirit.
5. Koblitz, Neal. (1984). "p-adic Numbers, p-adic Analysis, and Zeta-Functions". Springer-Verlag.