Occurrences of Grandi's series

Main article: Grandi's series

Binomial coefficients C (n, k) extended for negative and fractional n, illustrated with a simple binomial. It can be observed that Pascal's triangle is rotated and alternate terms are negated. The casen = −1 gives Grandi's series.

This article lists occurrences of the paradoxical infinite "sum" +1 -1 +1 -1 ... , sometimes called Grandi's series.

Parables

Further information: Grandi, Mathematical series analogy

Guido Grandi illustrated the series with a parable involving two brothers who share a gem.

Thomson's lamp is a supertask in which a hypothetical lamp is turned on and off infinitely many times in a finite time span. One can think of turning the lamp on as adding 1 to its state, and turning it off as subtracting 1. Instead of asking the sum of the series, one asks the final state of the lamp. ^[1]

One of the best-known classic parables to which infinite series have been applied, Achilles and the tortoise, can also be adapted to the case of Grandi's series. ^[2]

Numerical series

The Cauchy product of Grandi's series with itself is 1 − 2 + 3 − 4 + · · ·. ^[3]

Several series resulting from the introduction of zeros into Grandi's series have interesting properties; for these see Summation of Grandi's series#Dilution.

Grandi's series is just one example of a divergent geometric series.

The rearranged series 1 − 1 − 1 + 1 + 1 − 1 − 1 + · · · occurs in Euler's 1775 treatment of the pentagonal number theorem as the value of the Euler function at q = 1.

Power series

The power series most famously associated with Grandi's series is its ordinary generating function,

${\displaystyle f(x)=1-x+x^{2}-x^{3}+\cdots ={\frac {1}{1+x}}.}$

Fourier series

Hyperbolic sine

In his 1822 Théorie Analytique de la Chaleur, Joseph Fourier obtains what is currently called a Fourier sine series for a scaled version of the hyperbolic sine function,

${\displaystyle f(x)={\frac {\pi }{2\sinh \pi }}\sinh x.}$

He finds that the general coefficient of sin nx in the series is

${\displaystyle (-1)^{n-1}\left({\frac {1}{n}}-{\frac {1}{n^{3}}}+{\frac {1}{n^{5}}}-\cdots \right)=(-1)^{n-1}{\frac {n}{1+n^{2}}}.}$

For n > 1 the above series converges, while the coefficient of sin x appears as 1 − 1 + 1 − 1 + · · · and so is expected to be ¹⁄₂. In fact, this is correct, as can be demonstrated by directly calculating the Fourier coefficient from an integral:

${\displaystyle {\frac {2}{\pi }}\int _{0}^{\pi }f(x)\sin x\;dx=\left.{\frac {1}{2\sinh \pi }}(\cosh x\sin x-\sinh x\cos x)\right|_{0}^{\pi }={\frac {1}{2}}.}$ ^[4]

Dirac comb

Grandi's series occurs more directly in another important series,

${\displaystyle \cos x+\cos 2x+\cos 3x+\cdots =\sum _{k=1}^{\infty }\cos(kx).}$

At x = π, the series reduces to −1 + 1 − 1 + 1 − · · · and so one might expect it to meaningfully equal −¹⁄₂. In fact, Euler held that this series obeyed the formal relation Σ cos kx = −¹⁄₂, while d'Alembert rejected the relation, and Lagrange wondered if it could be defended by an extension of the geometric series similar to Euler's reasoning with Grandi's numerical series. ^[5]

Euler's claim suggests that

${\displaystyle 1+2\sum _{k=1}^{\infty }\cos(kx)=0?}$

for all x. This series is divergent everywhere, while its Cesàro sum is indeed 0 for almost all x. However, the series diverges to infinity at x = 2πn in a significant way: it is the Fourier series of a Dirac comb. The ordinary, Cesàro, and Abel sums of this series involve limits of the Dirichlet, Fejér, and Poisson kernels, respectively. ^[6]

Dirichlet series

Main article: Dirichlet eta function

Multiplying the terms of Grandi's series by 1/n^z yields the Dirichlet series

${\displaystyle \eta (z)=1-{\frac {1}{2^{z}}}+{\frac {1}{3^{z}}}-{\frac {1}{4^{z}}}+\cdots =\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n^{z}}},}$

which converges only for complex numbers z with a positive real part. Grandi's series is recovered by letting z = 0.

Unlike the geometric series, the Dirichlet series for η is not useful for determining what 1 − 1 + 1 − 1 + · · · "should" be. Even on the right half-plane, η(z) is not given by any elementary expression, and there is no immediate evidence of its limit as z approaches 0. ^[7] On the other hand, if one uses stronger methods of summability, then the Dirichlet series for η defines a function on the whole complex plane — the Dirichlet eta function — and moreover, this function is analytic. For z with real part > −1 it suffices to use Cesàro summation, and so η(0) = ¹⁄₂ after all.

The function η is related to a more famous Dirichlet series and function:

${\displaystyle {\begin{aligned}\eta (z)&=1+{\frac {1}{2^{z}}}+{\frac {1}{3^{z}}}+{\frac {1}{4^{z}}}+\cdots -{\frac {2}{2^{z}}}\left(1+{\frac {1}{2^{z}}}+\cdots \right)\\[5pt]&=\left(1-{\frac {2}{2^{z}}}\right)\zeta (z),\end{aligned}}}$

where ζ is the Riemann zeta function. Keeping Grandi's series in mind, this relation explains why ζ(0) = −¹⁄₂; see also 1 + 1 + 1 + 1 + · · ·. The relation also implies a much more important result. Since η(z) and (1 − 2^1−z) are both analytic on the entire plane and the latter function's only zero is a simple zero at z = 1, it follows that ζ(z) is meromorphic with only a simple pole at z = 1. ^[8]

Euler characteristics

Given a CW complex S containing one vertex, one edge, one face, and generally exactly one cell of every dimension, Euler's formula V − E + F − · · · for the Euler characteristic of S returns 1 − 1 + 1 − · · ·. There are a few motivations for defining a generalized Euler characteristic for such a space that turns out to be 1/2.

One approach comes from combinatorial geometry. The open interval (0, 1) has an Euler characteristic of −1, so its power set 2^{(0, 1)} should have an Euler characteristic of 2⁻¹ = 1/2. The appropriate power set to take is the "small power set" of finite subsets of the interval, which consists of the union of a point (the empty set), an open interval (the set of singletons), an open triangle, and so on. So the Euler characteristic of the small power set is 1 − 1 + 1 − · · ·. James Propp defines a regularized Euler measure for polyhedral sets that, in this example, replaces 1 − 1 + 1 − · · · with 1 − t + t² − · · ·, sums the series for |t| < 1, and analytically continues to t = 1, essentially finding the Abel sum of 1 − 1 + 1 − · · ·, which is 1/2. Generally, he finds χ(2^A) = 2^χ(A) for any polyhedral set A, and the base of the exponent generalizes to other sets as well. ^[9]

Infinite-dimensional real projective space RP^∞ is another structure with one cell of every dimension and therefore an Euler characteristic of 1 − 1 + 1 − · · ·. This space can be described as the quotient of the infinite-dimensional sphere by identifying each pair of antipodal points. Since the infinite-dimensional sphere is contractible, its Euler characteristic is 1, and its 2-to-1 quotient should have an Euler characteristic of 1/2. ^[10]

This description of RP^∞ also makes it the classifying space of Z₂, the cyclic group of order 2. Tom Leinster gives a definition of the Euler characteristic of any category which bypasses the classifying space and reduces to 1/|G| for any group when viewed as a one-object category. In this sense the Euler characteristic of Z₂ is itself ¹⁄₂. ^[11]

In physics

Grandi's series, and generalizations thereof, occur frequently in many branches of physics; most typically in the discussions of quantized fermion fields (for example, the chiral bag model), which have both positive and negative eigenvalues; although similar series occur also for bosons, such as in the Casimir effect.

The general series is discussed in greater detail in the article on spectral asymmetry, whereas methods used to sum it are discussed in the articles on regularization and, in particular, the zeta function regulator.

In art

The Grandi series has been applied to e.g. ballet by Benjamin Jarvis, in The Invariant journal. PDF here: https://invariants.org.uk/assets/TheInvariant_HT2016.pdf The noise artist Jliat has a 2000 musical single Still Life #7: The Grandi Series advertised as "conceptual art"; it consists of nearly an hour of silence. ^[12]

Notes

1. Rucker p.297
2. Saichev pp. 255–259
3. Hardy p.3
4. Bromwich p. 320
5. Ferraro 2005 p.17
6. Davis pp. 153–159
7. Knopp (p.458) makes this point to criticize Euler's use of analytical expressions to evaluate numerical series, saying "it need not at any rate be +¹⁄₂."
8. Knopp pp. 491–492
9. Propp pp. 7–8, 12
10. Propp, James (2002). "Euler measure as generalized cardinality". arXiv: math.CO/0203289 .
11. Leinster, Tom (2008). "The Euler characteristic of a category". Documenta Mathematica. 13: 21–49. arXiv: math/0610260 . Bibcode:2006math.....10260L. doi:10.4171/dm/240. S2CID   1046313.Baez, John (2006). "This Week's Finds in Mathematical Physics (Week 244)".
12. Review by George Zahora

References

• Bromwich, T.J. (1926) [1908]. An Introduction to the Theory of Infinite Series (2e ed.).
• Davis, Harry F. (May 1989). Fourier Series and Orthogonal Functions. Dover. ISBN   978-0-486-65973-2.
• Ferraro, Giovanni (2005). "Convergence and formal manipulation in the theory of series from 1730 to 1815". Historia Mathematica. 34: 62–88. doi: 10.1016/j.hm.2005.08.004 .
• Hardy, G.H. (1949). Divergent Series. Clarendon Press. LCC   QA295 .H29 1967.
• Knopp, Konrad (1990) [1922]. Theory and Application of Infinite Series. Dover. ISBN   978-0-486-66165-0.
• Propp, James (October 2003). "Exponentiation and Euler measure". Algebra Universalis. 29 (4): 459–471. arXiv: math.CO/0204009 . doi:10.1007/s00012-003-1817-1. S2CID   14340502.
• Rucker, Rudy (1995). Infinity and the mind: the science and philosophy of the infinite. Princeton UP. ISBN   978-0-691-00172-2.
• Saichev, A.I. & W.A. Woyczyński (1996). Distributions in the physical and engineering sciences, Volume 1. Birkhaüser. ISBN   978-0-8176-3924-2. LCC   QA324.W69 1996.