General considerations
Stability and linearity
The formal manipulations that lead to 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 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.[1]
Cesàro sum
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 σn 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
\( \sigma _{n}={\frac {1}{2}} \) for even n and \( \sigma _{n}={\frac {1}{2}}+{\frac {1}{2n}} \) for odd n.
This sequence of arithmetic means converges to 1⁄2, so the Cesàro sum of Σak is 1⁄2. Equivalently, one says that the Cesàro limit of the sequence 1, 0, 1, 0, … is 1⁄2.[2]
The Cesàro sum of 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.[3]
The series can also be summed by the more general fractional (C, a) methods.[4]
Abel sum
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,[5] and in practice he used it almost exclusively[6] 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
\( A\sum _{n=0}^{\infty }(-1)^{n}=\lim _{x\rightarrow 1}\sum _{n=0}^{\infty }(-x)^{n}=\lim _{x\rightarrow 1}{\frac {1}{1+x}}={\frac {1}{2}} \). [7]
Related series
The corresponding calculation that the Abel sum of 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.[8]
Dilution
Alternating spacing
That the ordinary Abel sum of 1 + 0 − 1 + 1 + 0 − 1 + · · · is 2⁄3 can also be phrased as the (A, λ) sum of the original series 1 − 1 + 1 − 1 + · · · where (λn) = (0, 2, 3, 5, 6, …). Likewise the (A, λ) sum of 1 − 1 + 1 − 1 + · · · where (λn) = (0, 1, 3, 4, 6, …) is 1⁄3.[9]
Power-law spacing
Exponential spacing
The summability of 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 22m−1 ≤ n ≤ 22m − 1 following a (− 1) term, the nth arithmetic means vary over the range
\( {\frac {2}{3}}\left({\frac {2^{2m}-1}{2^{2m}+2}}\right)\;\mathrm {to} \;{\frac {1}{3}}(1-2^{-2m}), \)
or about 2⁄3 to 1⁄3.[10]
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:
\( {\begin{array}{rcl}F(x)&=&\displaystyle x-x^{2}+x^{4}-x^{8}+\cdots \\[1em]&=&\displaystyle x-\left[(x^{2})-(x^{2})^{2}+(x^{2})^{4}-\cdots \right]\\[1em]&=&\displaystyle x-F(x^{2}).\end{array}} \)
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:
\( F(x)=\Psi (x)+\Phi (x) \)
where
\( \Phi (x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!(1+2^{n})}}\left(\log {\frac {1}{x}}\right)^{n} \)
satisfies the same functional equation as F. This now implies that 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.</ref> Since the Φ part has a limit of 1⁄2, F oscillates as well.
Separation of scales
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:
\( {\displaystyle S_{\varphi }=\lim _{\delta \downarrow 0}\sum _{k=0}^{\infty }(-1)^{k}\varphi (\delta k)={\frac {1}{2}}.} \)
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:
\( {\displaystyle {\begin{array}{rcl}S_{\varphi }&=&\displaystyle \lim _{\delta \downarrow 0}\sum _{k=0}^{\infty }\left[\varphi (2k\delta )-\varphi (2k\delta +\delta )\right]\\[1em]&=&\displaystyle \lim _{\delta \downarrow 0}\sum _{k=0}^{\infty }\varphi '(2k\delta +c_{k})(-\delta )\\[1em]&=&\displaystyle -{\frac {1}{2}}\int _{0}^{\infty }\varphi '(x)\,dx=-{\frac {1}{2}}\varphi (x)|_{0}^{\infty }={\frac {1}{2}}.\end{array}}} \) [11]
Euler transform and analytic continuation
Borel sum
The Borel sum of Grandi's series is again 1⁄2, since
\( 1-x+{\frac {x^{2}}{2!}}-{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}-\cdots =e^{-x} \)
and
\( \int _{0}^{\infty }e^{-x}e^{-x}\,dx=\int _{0}^{\infty }e^{-2x}\,dx={\frac {1}{2}} \).[12]
The series can also be summed by generalized (B, r) methods.[13]
Spectral asymmetry
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 { ω n } {\displaystyle \{\omega _{n}\}} \{\omega _{n}\} be a sequence of both positive and negative eigenvalues. Grandi's series corresponds to the formal sum
\( \sum _{n}\operatorname {sgn}(\omega _{n})\; \)
where \( \operatorname {sgn}(\omega _{n})=\pm 1 \) 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
\( \lim _{t\to 0}\sum _{n}\operatorname {sgn}(\omega _{n})e^{-t|\omega _{n}|} \)
which, for many interesting cases, is finite for non-zero t, and converges to a finite value in the limit.
Methods that fail
The integral function method with pn = exp (−cn2) and c > 0.[14]
The moment constant method with
\( d\chi =e^{-k(\log x)^{2}}x^{-1}dx \)
and k > 0.[15]
Geometric series
The geometric series in (x - 1),
\( {\displaystyle {\frac {1}{x}}=1-(x-1)+(x-1)^{2}-(x-1)^{3}+(x-1)^{4}-...} \)
is convergent for \( {\displaystyle |x-1|<1} \). Formally substituting x = 2 would give
\( {\displaystyle {\frac {1}{2}}=1-1+1-1+1-...} \)
However, x = 2 is outside the radius of convergence, \( {\displaystyle |x-1|<1} \), so this conclusion cannot be made.
Notes
Davis pp.152, 153, 157
Davis pp.153, 163
Davis pp.162-163, ex.1-5
Smail p.131
Kline 1983 p.313
Bromwich p.322
Davis p.159
Davis p.165
Hardy p.73
Hardy p.60
Saichev pp.260-262
Weidlich p.20
Smail p.128
Hardy pp.79-81, 85
Hardy pp.81-86
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.
Hardy, G.H. (1949). Divergent Series. Clarendon Press. LCC QA295 .H29 1967.
Kline, Morris (November 1983). "Euler and Infinite Series". Mathematics Magazine. 56 (5): 307–314. CiteSeerX 10.1.1.639.6923. doi:10.2307/2690371. JSTOR 2690371.
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.
Smail, Lloyd (1925). History and Synopsis of the Theory of Summable Infinite Processes. University of Oregon Press. LCC QA295 .S64.
Weidlich, John E. (June 1950). Summability methods for divergent series. Stanford M.S. theses.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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