Summation of Grandi's series#Separation of scales

The formal manipulations that lead to {{nowrap|1 − 1 + 1 − 1 + ⋯}} being assigned a value of ¹⁄₂ include:

• Adding or subtracting two series term-by-term,
• Multiplying through by a scalar term-by-term,
• "Shifting" the series with no change in the sum, and
• Increasing the sum by adding a new term to the series' head.

These are all legal manipulations for sums of convergent series, but {{nowrap|1 − 1 + 1 − 1 + ⋯}} is not a convergent series.

Nonetheless, there are many summation methods that respect these manipulations and that do assign a "sum" to Grandi's series. Two of the simplest methods are Cesàro summation and Abel summation.Davis pp.152, 153, 157

The first rigorous method for summing divergent series was published by Ernesto Cesàro in 1890. The basic idea is similar to Leibniz's probabilistic approach: essentially, the Cesàro sum of a series is the average of all of its partial sums. Formally one computes, for each n, the average σ_n of the first n partial sums, and takes the limit of these Cesàro means as n goes to infinity.

For Grandi's series, the sequence of arithmetic means is

:1, ¹⁄₂, ²⁄₃, ²⁄₄, ³⁄₅, ³⁄₆, ⁴⁄₇, ⁴⁄₈, …

or, more suggestively,

:(¹⁄₂+¹⁄₂), ¹⁄₂, (¹⁄₂+¹⁄₆), ¹⁄₂, (¹⁄₂+¹⁄₁₀), ¹⁄₂, (¹⁄₂+¹⁄₁₄), ¹⁄₂, …

where

:$\backslash sigma\_n=\backslash frac12$ for even n and $\backslash sigma\_n=\backslash frac12+\backslash frac\{1\}\{2n\}$ for odd n.

This sequence of arithmetic means converges to ¹⁄₂, so the Cesàro sum of Σa_k is ¹⁄₂. Equivalently, one says that the Cesàro limit of the sequence 1, 0, 1, 0, ⋯ is ¹⁄₂.Davis pp.153, 163

The Cesàro sum of {{nowrap|1 + 0 − 1 + 1 + 0 − 1 + ⋯}} is ²⁄₃. So the Cesàro sum of a series can be altered by inserting infinitely many 0s as well as infinitely many brackets.Davis pp.162-163, ex.1-5

The series can also be summed by the more general fractional (C, a) methods.Smail p.131

Abel summation is similar to Euler's attempted definition of sums of divergent series, but it avoids Callet's and N. Bernoulli's objections by precisely constructing the function to use. In fact, Euler likely meant to limit his definition to power series,Kline 1983 p.313 and in practice he used it almost exclusivelyBromwich p.322 in a form now known as Abel's method.

Given a series a₀ + a₁ + a₂ + ⋯, one forms a new series a₀ + a₁x + a₂x² + ⋯. If the latter series converges for 0 < x < 1 to a function with a limit as x tends to 1, then this limit is called the Abel sum of the original series, after Abel's theorem which guarantees that the procedure is consistent with ordinary summation. For Grandi's series one has

:$A\backslash sum\_\{n=0\}^\backslash infty(-1)^n\; =\; \backslash lim\_\{x\backslash rightarrow\; 1\}\backslash sum\_\{n=0\}^\backslash infty(-x)^n\; =\; \backslash lim\_\{x\backslash rightarrow\; 1\}\backslash frac\{1\}\{1+x\}=\backslash frac12.$ Davis p.159

The corresponding calculation that the Abel sum of {{nowrap|1 + 0 − 1 + 1 + 0 − 1 + ⋯}} is ²⁄₃ involves the function (1 + x)/(1 + x + x²).

Whenever a series is Cesàro summable, it is also Abel summable and has the same sum. On the other hand, taking the Cauchy product of Grandi's series with itself yields a series which is Abel summable but not Cesàro summable: 1 − 2 + 3 − 4 + ⋯ has Abel sum ¹⁄₄.Davis p.165

That the ordinary Abel sum of {{nowrap|1 + 0 − 1 + 1 + 0 − 1 + ⋯}} is ²⁄₃ can also be phrased as the (A, λ) sum of the original series {{nowrap|1 − 1 + 1 − 1 + ⋯}} where (λ_n) = (0, 2, 3, 5, 6, ...). Likewise the (A, λ) sum of {{nowrap|1 − 1 + 1 − 1 + ⋯}} where (λ_n) = (0, 1, 3, 4, 6, ...) is ¹⁄₃.Hardy p.73

The summability of {{nowrap|1 − 1 + 1 − 1 + ⋯}} can be frustrated by separating its terms with exponentially longer and longer groups of zeros. The simplest example to describe is the series where (−1)ⁿ appears in the rank 2ⁿ:

:0 + 1 − 1 + 0 + 1 + 0 + 0 + 0 − 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 1 + 0 + ⋯.

This series is not Cesaro summable. After each nonzero term, the partial sums spend enough time lingering at either 0 or 1 to bring the average partial sum halfway to that point from its previous value. Over the interval {{nowrap|2^2m−1 ≤ n ≤ 2^2m − 1}} following a (− 1) term, the nth arithmetic means vary over the range

:$\backslash frac\{2\}\{3\}\; \backslash left(\backslash frac\{2^\{2m\}-1\}\{2^\{2m\}+2\}\backslash right)\backslash ;\backslash mathrm\{to\}\backslash ;\backslash frac\{1\}\{3\}(1-2^\{-2m\}),$

or about ²⁄₃ to ¹⁄₃.Hardy p.60

In fact, the exponentially spaced series is not Abel summable either. Its Abel sum is the limit as x approaches 1 of the function

:F(x) = 0 + x − x² + 0 + x⁴ + 0 + 0 + 0 − x⁸ + 0 + 0 + 0 + 0 + 0 + 0 + 0 + x¹⁶ + 0 + ⋯.

This function satisfies a functional equation:

:$\backslash begin\{array\}\{rcl\}$

F(x) & = &\displaystyle x-x^2+x^4-x^8+\cdots \\[1em]

& = & \displaystyle x - \left[(x^2)-(x^2)^2+(x^2)^4-\cdots\right] \\[1em]

& = & \displaystyle x-F(x^2).

\end{array}

This functional equation implies that F(x) roughly oscillates around ¹⁄₂ as x approaches 1. To prove that the amplitude of oscillation is nonzero, it helps to separate F into an exactly periodic and an aperiodic part:

:$F(x)\; =\; \backslash Psi(x)\; +\; \backslash Phi(x)$

where

:$\backslash Phi(x)\; =\; \backslash sum\_\{n=0\}^\backslash infty\backslash frac\{(-1)^n\}\{n!(1+2^n)\}\backslash left(\backslash log\backslash frac\; 1x\backslash right)^n$

satisfies the same functional equation as F. This now implies that {{nowrap|1=Ψ(x) = −Ψ(x²) = Ψ(x⁴)}}, so Ψ is a periodic function of loglog(1/x). Since dy (p.77) speaks of "another solution" and "plainly not constant", although technically he does not prove that F and Φ are different. Since the Φ part has a limit of ¹⁄₂, F oscillates as well.

Given any function φ(x) such that φ(0) = 1, and the derivative of φ is integrable over (0, +∞), then the generalized φ-sum of Grandi's series exists and is equal to ¹⁄₂:

:$S\_\backslash varphi\; =\; \backslash lim\_\{\backslash delta\backslash downarrow0\}\backslash sum\_\{k=0\}^\backslash infty(-1)^k\backslash varphi(\backslash delta\; k)\; =\; \backslash frac12.$

The Cesaro or Abel sum is recovered by letting φ be a triangular or exponential function, respectively. If φ is additionally assumed to be continuously differentiable, then the claim can be proved by applying the mean value theorem and converting the sum into an integral. Briefly:

:$\backslash begin\{array\}\{rcl\}$

S_\varphi & = &\displaystyle \lim_{\delta\downarrow0}\sum_{k=0}^\infty\left[\varphi(2k\delta) - \varphi(2k\delta+\delta)\right] \\[1em]

& = & \displaystyle \lim_{\delta\downarrow0}\sum_{k=0}^\infty\varphi'(2k\delta+c_k)(-\delta) \\[1em]

& = & \displaystyle-\frac12\int_0^\infty\varphi'(x) \,dx = -\frac12\varphi(x)|_0^\infty = \frac12.

\end{array}Saichev pp.260-262

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The Borel sum of Grandi's series is again ¹⁄₂, since

:$1-x+\backslash frac\{x^2\}\{2!\}-\backslash frac\{x^3\}\{3!\}+\backslash frac\{x^4\}\{4!\}-\backslash cdots=e^\{-x\}$

and

:$\backslash int\_0^\backslash infty\; e^\{-x\}e^\{-x\}\backslash ,dx=\backslash int\_0^\backslash infty\; e^\{-2x\}\backslash ,dx=\backslash frac12.$Weidlich p.20

The series can also be summed by generalized (B, r) methods.Smail p.128

The entries in Grandi's series can be paired to the eigenvalues of an infinite-dimensional operator on Hilbert space. Giving the series this interpretation gives rise to the idea of spectral asymmetry, which occurs widely in physics. The value that the series sums to depends on the asymptotic behaviour of the eigenvalues of the operator. Thus, for example, let $\backslash \{\backslash omega\_n\backslash \}$ be a sequence of both positive and negative eigenvalues. Grandi's series corresponds to the formal sum

:$\backslash sum\_n\; \backslash sgn(\backslash omega\_n)\backslash ;$

where $\backslash sgn(\backslash omega\_n)=\backslash pm\; 1$ is the sign of the eigenvalue. The series can be given concrete values by considering various limits. For example, the heat kernel regulator leads to the sum

:$\backslash lim\_\{t\backslash to\; 0\}\; \backslash sum\_n\; \backslash sgn(\backslash omega\_n)\; e^\{-t|\backslash omega\_n|\}$

which, for many interesting cases, is finite for non-zero t, and converges to a finite value in the limit.

The integral function method with p_n = exp (−cn²) and c > 0.Hardy pp.79-81, 85

The moment constant method with

:$d\backslash chi\; =\; e^\{-k(\backslash log\; x)^2\}x^\{-1\}dx$

and k > 0.Hardy pp.81-86

The geometric series in $(x\; -\; 1)$,

$\backslash frac\{1\}\{x\}\; =\; 1\; -\; (x-1)\; +\; (x-1)^2\; -\; (x-1)^3\; +\; (x-1)^4\; -\; ..\; .$

is convergent for . Formally substituting $x\; =\; 2$ would give

$\backslash frac\{1\}\{2\}\; =\; 1\; -\; 1\; +\; 1\; -\; 1\; +\; 1\; -\; ...$

However, $x\; =\; 2$ is outside the radius of convergence, , so this conclusion cannot be made.

{{reflist|20em}}

{{refbegin}}

• {{cite book |last=Bromwich |first=T.J. |year=1926 |origyear=1908 |edition=2e |title=An Introduction to the Theory of Infinite Series}}
• {{cite book |last=Davis |first=Harry F. |title=Fourier Series and Orthogonal Functions |date=May 1989 |publisher=Dover |isbn=978-0-486-65973-2}}
• {{cite book |last=Hardy |first=G.H. |authorlink=G. H. Hardy |title=Divergent Series |year=1949 |publisher=Clarendon Press |id={{LCC|QA295|.H29|1967}}}}
• {{cite journal |doi=10.2307/2690371 |last=Kline |first=Morris |title=Euler and Infinite Series |journal=Mathematics Magazine |volume=56 |issue=5 |date=November 1983 |pages=307–314 |jstor=2690371|citeseerx=10.1.1.639.6923 }}
• {{cite book |author1=Saichev, A.I. |author2=W.A. Woyczyński |name-list-style=amp |title=Distributions in the physical and engineering sciences, Volume 1 |publisher=Birkhaüser |year=1996 |isbn=978-0-8176-3924-2|id={{LCC|QA324.W69|1996}}}}
• {{cite book |last=Smail |first=Lloyd |title=History and Synopsis of the Theory of Summable Infinite Processes |year=1925 |publisher=University of Oregon Press |id={{LCC|QA295|.S64}}}}
• {{cite book |author=Weidlich |first=John E. |title=Summability methods for divergent series |date=June 1950 |publisher=Stanford M.S. theses}}

{{refend}}

{{Grandi's series}}

Grandi's series, Summation of

Category:Grandi's series