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
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:
\begin{array}{rcl}
s&=&\displaystyle1+2+4+8+16+ … \\
&=&\displaystyle1+2(1+2+4+8+ … )\\
&=&\displaystyle1+2s
\end{array}
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
References
- Euler . Leonhard . Leonhard Euler . De seriebus divergentibus . Novi Commentarii Academiae Scientiarum Petropolitanae . 5 . 1760 . 205–237 .
- Book: Gardiner, A. . Anthony Gardiner (mathematician) . Understanding infinity: the mathematics of infinite processes . 2002 . 1982 . Dover . Dover . 0-486-42538-X.
- Book: Hardy, G. H. . G. H. Hardy . Divergent Series . 1949 . Clarendon Press . .
Further reading
- Barbeau . E. J. . P. J.. Leah . Euler's 1760 paper on divergent series . Historia Mathematica . 3 . 2 . 141–160 . 10.1016/0315-0860(76)90030-6. May 1976. free .
- Ferraro . Giovanni . Convergence and Formal Manipulation of Series from the Origins of Calculus to About 1730 . Annals of Science . 59 . 2002 . 2 . 179–199 . 10.1080/00033790010028179. 143992318 .
- Kline . Morris . Morris Kline . Euler and Infinite Series . Mathematics Magazine . 56 . 5 . 307–314 . 10.2307/2690371 . 2690371. November 1983.
- Web site: Sandifer . Ed . Divergent series . How Euler Did It . MAA Online . June 2006 . 2007-02-17 . 2013-03-20 . https://web.archive.org/web/20130320011704/http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2032%20divergent%20series.pdf . dead .
- Sierpińska . Anna . Anna Sierpińska. Humanities students and epistemological obstacles related to limits . Educational Studies in Mathematics . 18 . 4 . 371–396 . 10.1007/BF00240986 . 3482354. November 1987. 144880659 .
Notes and References
- Hardy p. 10
- Hardy pp. 8, 10
- The two roots of
are briefly touched on by Hardy p. 19.
- Gardiner pp. 93–99; the argument on p. 95 for
is slightly different but has the same spirit.
- 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.