Summation of Grandi's series
General considerations. Stability and linearity. The formal manipulations that lead to 1 − 1 + 1 − 1 + · · · being assigned a value of ⁄ include: 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. 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 σ 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 or, more suggestively, where This sequence of arithmetic means converges to ⁄, so the Cesàro sum of Σ"a" is ⁄. Equivalently, one says that the Cesàro limit of the sequence 1, 0, 1 , 0, … is ⁄. The Cesàro sum of 1 + 0 − 1 + 1 + 0 − 1 + · · · is ⁄. So the Cesàro sum of a series can be altered by inserting infinitely many 0s as well as infinitely many brackets. The series can also be summed by the more general fractional (C, a) methods. 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, and in practice he used it almost exclusively in a form now known as Abel's method. Given a series "a" + "a" + "a" + · · ·, one forms a new series "a" + "a""x" + "a""x" + · · ·. 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 Related series. The corresponding calculation that the Abel  sum of 1 + 0 − 1 + 1 + 0 − 1 + · · · is ⁄ involves the function (1 + "x")/(1 + "x" + "x"). 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: has Abel sum ⁄. Dilution. Alternating spacing. That the ordinary Abel sum of 1 + 0 − 1 + 1 + 0 − 1 + · · · is ⁄ can also be phrased as the  (A, λ) sum of the original series 1 − 1 + 1 − 1 + · · · where (λ) = (0, 2, 3, 5, 6, …). Likewise the (A, λ) sum of 1 − 1 + 1 − 1 + · · · where (λ) = (0, 1, 3, 4, 6, …) is ⁄. 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) appears in the rank 2: 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 following a (− 1) term, the "n"th arithmetic means vary over the range or about ⁄ to ⁄. In fact, the exponentially spaced series is not Abel summable either. Its Abel sum is the limit as "x" approaches 1 of the function This function satisfies a functional equation: This functional equation implies that "F"("x") roughly oscillates around ⁄ 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: where satisfies the same functional equation as "F". This now implies that , 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 ⁄, "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 ⁄: 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: Borel sum. The Borel sum of Grandi's series is again ⁄, since and The series can also be summed by generalized (B, r) methods. 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 formula_12 be a sequence of both positive and negative eigenvalues. Grandi's series corresponds to the formal sum where formula_14 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 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 "p" = exp (−"cn") and "c" > 0. The moment constant method with and "k" > 0. Geometric series. The geometric series in formula_17, formula_18 is convergent for formula_19. Formally substituting formula_20 would give formula_21 However, formula_20 is outside the radius of convergence, formula_19, so this conclusion cannot be made.