1 + 2 + 4 + 8 + ⋯ Explained

In mathematics, 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.

Summation

The partial sums of

1+2+4+8+

are

1,3,7,15,\ldots;

since these diverge to infinity, so does the series. 2^0+2^1 + \cdots + 2^k = 2^-1

It is written as

infty
\sum
n=0

2n

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

1+2+4+8+

to the finite value of −1. The associated power seriesf(x) = 1 + 2x + 4x^2 + 8x^3+ \cdots + 2^nx^n + \cdots = \frac has a radius of convergence around 0 of only
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=

1
2
deleted, and it is given by the same rule

f(x)=

1
1-2x

.

Since

f(1)=-1,

the original series

1+2+4+8+

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,1 + y + y^2 + y^3 + \cdots = \fracand plugging in

y=2.

These two series are related by the substitution

y=2x.

The fact that (E) summation assigns a finite value to

1+2+4+8+

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{array}{rcl} s&=&\displaystyle1+2+4+8+16+ … \\ &=&\displaystyle1+2(1+2+4+8+ … )\\ &=&\displaystyle1+2s \end{array}

In a useful sense,

s=infty

is a root of the equation

s=1+2s.

(For example,

infty

is one of the two fixed points of the Möbius transformation

z\mapsto1+2z

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.

[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

1-1+1-1+

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

1
9
and

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]

See also

References

Further reading

Notes and References

  1. Hardy p. 10
  2. Hardy pp. 8, 10
  3. The two roots of

    s=1+2s

    are briefly touched on by Hardy p. 19.
  4. Gardiner pp. 93–99; the argument on p. 95 for

    1+2+4+8+

    is slightly different but has the same spirit.
  5. Book: Koblitz, Neal. p-adic Numbers, p-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics, vol. 58. Springer-Verlag. 0-387-96017-1. 1984. chapter I, exercise 16, p. 20.