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.

since these diverge to infinity, so does the series.

It is written as 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

Nonetheless, the so-defined function

has a unique analytic continuation to the complex plane with the point

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,

These two series are related by the substitution

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,

is one of the two fixed points of the Möbius transformation

If some summation method is known to return an ordinary number for

may be subtracted from both sides of the equation, yielding

[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

(Grandi's series), where a series of integers appears to have the non-integer sum

These examples illustrate the potential danger in applying similar arguments to the series implied by such recurring decimals as

The arguments are ultimately justified for these convergent series, implying that

but the underlying proofs demand careful thinking about the interpretation of endless sums.

As a series of 2-adic numbers this series converges to the same sum, −1, as was derived above by analytic continuation.