General considerations Stability and linearity The formal manipulations that lead to 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 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
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 : \sigma_n=\frac12 for even
n and \sigma_n=\frac12+\frac{1}{2n} for odd
n. This sequence of arithmetic means converges to 1⁄2, so the Cesàro sum of Σ
ak is 1⁄2. Equivalently, one says that the Cesàro limit of the sequence 1, 0, 1, 0, ⋯ is 1⁄2. The Cesàro sum of is 2⁄3. 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
a0 +
a1 +
a2 + ⋯, one forms a new series
a0 +
a1
x +
a2
x2 + ⋯. If the latter series converges for 0 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}=\frac12.
Related series The corresponding calculation that the Abel sum of 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.
Dilution Alternating spacing That the ordinary Abel sum of is 2⁄3 can also be phrased as the (A, λ) sum of the original series where (λ
n) = (0, 2, 3, 5, 6, ...). Likewise the (A, λ) sum of where (λ
n) = (0, 1, 3, 4, 6, ...) is 1⁄3.
Power-law spacing Exponential spacing The summability of 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 2
n: : 0 + 1 − 1 + 0 + 1 + 0 + 0 + 0 − 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 1 + 0 + ⋯. This series is not Cesàro 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
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. 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 1x\right)^n 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. 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: : S_\varphi = \lim_{\delta\downarrow0}\sum_{k=0}^\infty(-1)^k\varphi(\delta k) = \frac12. The Cesàro 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: : \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}
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=\frac12. 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 \{\omega_n\} be a sequence of both positive and negative eigenvalues. Grandi's series corresponds to the formal sum : \sum_n \sgn(\omega_n)\; where \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 \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. The
moment constant method with : d\chi = e^{-k(\log x)^2}x^{-1}dx and
k > 0.
Geometric series The
geometric series in (x - 1), \frac{1}{x} = 1 - (x-1) + (x-1)^2 - (x-1)^3 + (x-1)^4 - .. . is convergent for |x - 1| .
Formally substituting x = 2 would give \frac{1}{2} = 1 - 1 + 1 - 1 + 1 - ... However, x = 2 is outside the
radius of convergence, |x - 1| , so this conclusion cannot be made. ==Related problems==