The formal manipulations that lead to 1 − 1 + 1 − 1 + ⋯ being assigned a value of 1⁄2 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.
[1] 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 or, more suggestively, where This sequence of arithmetic means converges to 1⁄2, so the Cesàro sum of Σak is 1⁄2.
[4] 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.
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 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).
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.
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 or about 2⁄3 to 1⁄3.
[10] In fact, the exponentially spaced series is not Abel summable either.
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.
[13] The entries in Grandi's series can be paired to the eigenvalues of an infinite-dimensional operator on Hilbert space.
The value that the series sums to depends on the asymptotic behaviour of the eigenvalues of the operator.
Grandi's series corresponds to the formal sum where
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.