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.

The first four partial sums of 1 + 2 + 4 + 8 + ⋯.

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

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The partial sums of   are   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   to the finite value of −1. The associated power series   has a radius of convergence around 0 of only   so it does not converge at   Nonetheless, the so-defined function   has a unique analytic continuation to the complex plane 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 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,   and plugging in   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 a root of the equation   (For example,   is one of the two fixed points of the Möbius transformation   on the Riemann sphere). If some summation method is known to return an ordinary number for  ; that is, not   then it is easily determined. In this case   may be subtracted from both sides of the equation, yielding   so  [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   and most notably  . The arguments are ultimately justified for these convergent series, implying that   and   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]

See also

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Notes

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  1. ^ Hardy p. 10
  2. ^ Hardy pp. 8, 10
  3. ^ The two roots of   are briefly touched on by Hardy p. 19.
  4. ^ Gardiner pp. 93–99; the argument on p. 95 for   is slightly different but has the same spirit.
  5. ^ 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

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Further reading

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