Summation of Grandi's series

General considerations

Stability and linearity

The formal manipulations that lead to being assigned a value of ¹Ã¢ÂÂ„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, ¹Ã¢ÂÂ„₂, ²Ã¢ÂÂ„₃, ²Ã¢ÂÂ„₄, ³Ã¢ÂÂ„₅, ³Ã¢ÂÂ„₆, ⁴⁄₇, ⁴⁄₈, …

or, more suggestively,

(¹Ã¢ÂÂ„₂+¹Ã¢ÂÂ„₂), ¹Ã¢ÂÂ„₂, (¹Ã¢ÂÂ„₂+¹Ã¢ÂÂ„₆), ¹Ã¢ÂÂ„₂, (¹Ã¢ÂÂ„₂+¹Ã¢ÂÂ„₁₀), ¹Ã¢ÂÂ„₂, (¹Ã¢ÂÂ„₂+¹Ã¢ÂÂ„₁₄), ¹Ã¢ÂÂ„₂, …

where

for even n and for odd n.

This sequence of arithmetic means converges to ¹Ã¢ÂÂ„₂, so the Cesàro sum of Σa_k is ¹Ã¢ÂÂ„₂. Equivalently, one says that the Cesàro limit of the sequence 1, 0, 1, 0, ⋯ is ¹Ã¢ÂÂ„₂.

The Cesàro sum of is ²Ã¢ÂÂ„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 + 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

Related series

The corresponding calculation that the Abel sum of is ²Ã¢ÂÂ„₃ involves the function (1&nbsp;+&nbsp;x)/(1&nbsp;+&nbsp;x&nbsp;+&nbsp;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 ¹Ã¢ÂÂ„₄.

Dilution

Alternating spacing

That the ordinary Abel sum of is ²Ã¢ÂÂ„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 ¹Ã¢ÂÂ„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 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 following a (− 1) term, the nth arithmetic means vary over the range

or about ²Ã¢ÂÂ„₃ to ¹Ã¢ÂÂ„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:

This functional equation implies that F(x) roughly oscillates around ¹Ã¢ÂÂ„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:

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. Since the Φ part has a limit of ¹Ã¢ÂÂ„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 ¹Ã¢ÂÂ„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:

Euler transform and analytic continuation

Borel sum

The Borel sum of Grandi's series is again ¹Ã¢ÂÂ„₂, 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 be a sequence of both positive and negative eigenvalues. Grandi's series corresponds to the formal sum

where 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_n = exp (&minus;cn²) and c > 0.

The moment constant method with

and k > 0.

Geometric series

The geometric series in ,

is convergent for . Formally substituting would give

However, is outside the radius of convergence, , so this conclusion cannot be made.

Notes

References