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| {{main|Grandi's series}}
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| ==Parables==
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| {{further2|[[History of Grandi's series|Grandi]], [[Thomson's lamp|Mathematical series analogy]]}}
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| [[Guido Grandi]] illustrated the series with a parable involving two brothers who share a gem.
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| [[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.<ref>Rucker p.297</ref> | |
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| 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.<ref>Saichev pp. 255–259</ref>
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| ==Numerical series==
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| The [[Cauchy product]] of Grandi's series with itself is [[1 − 2 + 3 − 4 + · · ·]].<ref>Hardy p.3</ref>
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| Several series resulting from the introduction of zeros into Grandi's series have interesting properties; for these see [[Summation of Grandi's series#Dilution]].
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| Grandi's series is just one example of a [[divergent geometric series]].
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| 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.
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| ==Power series==
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| The power series most famously associated with Grandi's series is its [[ordinary generating function]],
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| :<math>f(x) = 1-x+x^2-x^3+\cdots = \frac{1}{1+x}.</math>
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| ==Fourier series==
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| ===Hyperbolic sine===
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| In his 1822 ''Théorie Analytique de la Chaleur'', [[Joseph Fourier]] obtains what we now call a [[Fourier sine series]] for a scaled version of the [[hyperbolic sine]] function,
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| :<math>f(x) = \frac{\pi}{2\sinh\pi} \sinh x.</math>
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| He finds that the general coefficient of sin ''nx'' in the series is
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| :<math>(-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}.</math>
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| 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 <sup>1</sup>⁄<sub>2</sub>. In fact, this is correct, as can be demonstrated by directly calculating the Fourier coefficient from an integral:
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| :<math>\frac 2 \pi \int_0^\pi f(x)\sin x \;dx = \frac{1}{2\sinh\pi}\left.(\cosh x \sin x - \sinh x \cos x)\right|_0^\pi = \frac 1 2.</math><ref>Bromwich p. 320</ref>
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| ===Dirac comb===
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| Grandi's series occurs more directly in another important series,
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| :<math>\cos x + \cos 2x + \cos 3x + \cdots = \sum_{k=1}^\infty\cos(kx).</math>
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| At ''x'' = π, the series reduces to −1 + 1 − 1 + 1 − · · · and so one might expect it to meaningfully equal −<sup>1</sup>⁄<sub>2</sub>. In fact, Euler held that this series obeyed the formal relation Σ cos ''kx'' = −<sup>1</sup>⁄<sub>2</sub>, 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.<ref>Ferraro 2005 p.17</ref>
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| Euler's claim suggests that
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| :<math>1 +2\sum_{k=1}^\infty\cos(kx) = 0?</math>
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| 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'' = 2π''n'' 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 kernel|Dirichlet]], [[Fejér kernel|Fejér]], and [[Poisson kernel]]s, respectively.<ref>Davis pp. 153–159</ref>
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| ==Dirichlet series==
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| {{main|Dirichlet eta function}}
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| Multiplying the terms of Grandi's series by 1/''n''<sup>''z''</sup> yields the [[Dirichlet series]]
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| :<math>\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},</math> | |
| which converges only for complex numbers ''z'' with a positive real part. Grandi's series is recovered by letting ''z'' = 0.
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| 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.<ref>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 +<sup>1</sup>⁄<sub>2</sub>."</ref> 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 function|analytic]]. For ''z'' with real part > −1 it suffices to use Cesàro summation, and so η(0) = <sup>1</sup>⁄<sub>2</sub> after all.
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| The function η is related to a more famous Dirichlet series and function:
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| :<math>\begin{array}{rcl}
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| \eta(z) & = &\displaystyle 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) \\[1em]
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| & = & \displaystyle \left(1-\frac{2}{2^z}\right)\zeta(z),
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| \end{array}</math>
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| where ζ is the [[Riemann zeta function]]. Keeping Grandi's series in mind, this relation explains why ζ(0) = −<sup>1</sup>⁄<sub>2</sub>; see also [[1 + 1 + 1 + 1 + · · ·]]. The relation also implies a much more important result. Since η(''z'') and (1 − 2<sup>1−''z''</sup>) are both analytic on the entire plane and the latter function's only [[zero (complex analysis)|zero]] is a [[simple zero]] at ''z'' = 1, it follows that ζ(''z'') is [[meromorphic function|meromorphic]] with only a [[simple pole]] at ''z'' = 1.<ref>Knopp pp. 491–492</ref>
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| ==Euler characteristics==
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| Given a [[CW complex]] ''S'' containing one vertex, one edge, one face, and generally exactly one cell of every dimension, Euler's formula {{nowrap|''V'' − ''E'' + ''F'' − · · ·}} for the [[Euler characteristic]] of ''S'' returns {{nowrap|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.
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| One approach comes from [[combinatorial geometry]]. The open interval (0, 1) has an Euler characteristic of −1, so its power set 2<sup>(0, 1)</sup> should have an Euler characteristic of 2<sup>−1</sup> = 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 singetons), an open triangle, and so on. So the Euler characteristic of the small power set is {{nowrap|1 − 1 + 1 − · · ·}}. [[James Propp]] defines a regularized Euler measure for [[polyhedral set]]s that, in this example, replaces {{nowrap|1 − 1 + 1 − · · ·}} with {{nowrap|1 − ''t'' + ''t''<sup>2</sup> − · · ·}}, sums the series for |''t''| < 1, and analytically continues to ''t'' = 1, essentially finding the Abel sum of {{nowrap|1 − 1 + 1 − · · ·}}, which is 1/2. Generally, he finds χ(2<sup>''A''</sup>) = 2<sup>χ(''A'')</sup> for any polyhedral set ''A'', and the base of the exponent generalizes to other sets as well.<ref>Propp pp. 7–8, 12</ref>
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| [[Infinite-dimensional real projective space|Infinite-dimensional]] [[real projective space]] '''RP'''<sup>∞</sup> is another structure with one cell of every dimension and therefore an Euler characteristic of {{nowrap|1 − 1 + 1 − · · ·}}. This space can be described as the quotient of the [[infinite-dimensional sphere|infinite-dimensional]] [[sphere]] by identifying each pair of [[antipodal point]]s. 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.<ref>{{cite arxiv |first=James |last=Propp |title=Euler measure as generalized cardinality |year=2002 |eprint=math.CO/0203289 |class=math.CO}}</ref>
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| This description of '''RP'''<sup>∞</sup> also makes it the [[classifying space]] of Z<sub>2</sub>, the [[cyclic group]] [[cyclic group of order 2|of order 2]]. Tom Leinster gives a definition of the Euler characteristic of any [[category (mathematics)|category]] which bypasses the classifying space and reduces to 1/|''G''| for any [[group (mathematics)|group]] when viewed as a one-object category. In this sense the Euler characteristic of Z<sub>2</sub> is itself <sup>1</sup>⁄<sub>2</sub>.<ref>{{cite journal |last=Leinster |first=Tom |title=The Euler characteristic of a category |year=2006 |publisher=[[arXiv]] |pages=21–49 |volume=13 |journal=Documenta Mathematica |arxiv=math/0610260}} {{cite web |last=Baez |first=John |title= This Week's Finds in Mathematical Physics (Week 244) |year=2006 |url=http://math.ucr.edu/home/baez/week244.html}}</ref>
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| == 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 [[eigenvalue]]s; although similar series occur also for [[boson]]s, such as in the [[Casimir effect]].
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| 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 (physics)|regularization]] and, in particular, the [[zeta function regulator]].
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| ==In art==
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| [[Jliat]]'s 2000 musical single ''Still Life #7: The Grandi Series'' advertises itself as "conceptual art"; it consists of nearly an hour of silence.<ref>[http://www.splendidezine.com/reviews/may-7-01/aag.html Review by George Zahora]</ref>
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| ==Notes==
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| {{Reflist|2}}
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| ==References==
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| <div class="references-small"> | |
| *{{cite book |last=Bromwich |first=T.J. |year=1926 |origyear=1908 |edition=2e |title=An Introduction to the Theory of Infinite Series}}
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| *{{cite book |last=Davis |first=Harry F. |title=Fourier Series and Orthogonal Functions |date=May 1989 |publisher=Dover |isbn=0-486-65973-9}}
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| *{{cite journal |last=Ferraro |first=Giovanni |title=Convergence and formal manipulation in the theory of series from 1730 to 1815 |journal=Historia Mathematica |year=2005 |doi=10.1016/j.hm.2005.08.004 |volume=34 |pages=62}}
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| *{{cite book |last=Hardy |first=G.H. |authorlink=G. H. Hardy |title=Divergent Series |year=1949 |publisher=Clarendon Press |id={{LCC|QA295|.H29|1967}}}}
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| *{{cite book |last=Knopp |first=Konrad |authorlink=Konrad Knopp |title=Theory and Application of Infinite Series |year=1990 |origyear=1922 |publisher=Dover |isbn=0-486-66165-2}}
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| *{{cite journal |last=Propp |first=James |title=Exponentiation and Euler measure |journal=Algebra Universalis |volume=29 |issue=4 |date=October 2003 |pages=459–471 |doi=10.1007/s00012-003-1817-1 |arxiv=math.CO/0204009}}
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| *{{cite book |last=Rucker |first=Rudy |title=Infinity and the mind: the science and philosophy of the infinite |year=1995 |publisher=Princeton UP |isbn=0-691-00172-3}}
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| *{{cite book |author=Saichev, A.I., and W.A. Woyczyński |title=Distributions in the physical and engineering sciences, Volume 1 |publisher=Birkhaüser |year=1996 |isbn=0-8176-3924-1 | id={{LCC|QA324.W69|1996}}}}
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| </div>
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| [[Category:Divergent series|Grandi's series, Occurrences of]]
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