Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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 number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s it diverges
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit....
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of 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.
The partial sums of 1 + 2 + 4 + 8 + … are since these diverge to infinity, so does the series. Therefore any totally regular summation method gives a sum of infinity, including the Cesàro sum
In mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. If the series converges in the usual sense to a sum A, then the series is also Cesàro summable and has Cesàro sum A...
and Abel sum.
On the other hand, there is at least one generally useful method that sums to the finite value of −1. The associated power series
has a radius of convergence
In mathematics, the radius of convergence of a power series is a quantity, either a non-negative real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well...
of only 1
, so it does not converge at . Nonetheless, the so-defined function f
has a unique analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
to the complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
with the point deleted, and it is given by the same rule . Since , the original series 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
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
in reference to Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
's approach to divergent series.)
An almost identical approach is to consider the power series whose coefficients are all 1, i.e.
and plugging in y
= 2. Of course these two series are related by the substitution y
The fact that (E
) summation assigns a finite value to 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:
In a useful sense, s
= ∞ is a root of the equation (For example, ∞ is one of the two fixed point
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
s of the Möbius transformation
on the Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
.) If some summation method is known to return an ordinary number for s
not ∞, then it is easily determined. In this case s
may be subtracted from both sides of the equation, yielding , so .
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 integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s appears to have the non-integer sum 1
. 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…
In mathematics, the repeating decimal 0.999... denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number...
. The arguments are ultimately justified for these convergent series, implying that and , but the underlying proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
s 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
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
. As a series of 2-adic numbers this series converges to the same sum, −1, as was derived above by analytic continuation.