Les séries divergentes sont en général quelque chose de bien fatal et c’est une honte qu’on ose y fonder aucune démonstration.
For example, Cesàro summation assigns Grandi's divergent series the value 1/2.
Cesàro summation is an averaging method, in that it relies on the arithmetic mean of the sequence of partial sums.
In physics, there are a wide variety of summability methods; these are discussed in greater detail in the article on regularization.
Before the 19th century, divergent series were widely used by Leonhard Euler and others, but often led to confusing and contradictory results.
A summability method M is regular if it agrees with the actual limit on all convergent series.
More subtle, are partial converse results, called Tauberian theorems, from a prototype proved by Alfred Tauber.
This fact is not very useful in practice, since there are many such extensions, inconsistent with each other, and also since proving such operators exist requires invoking the axiom of choice or its equivalents, such as Zorn's lemma.
The advent of Wiener's tauberian theorem marked an epoch in the subject, introducing unexpected connections to Banach algebra methods in Fourier analysis.
Summation of divergent series is also related to extrapolation methods and sequence transformations as numerical techniques.
A summation method can be seen as a function from a set of sequences of partial sums to values.
There are certain properties it is desirable for these methods to possess if they are to arrive at values corresponding to limits and sums, respectively.
(Using this language, a summation method A is regular iff it is consistent with the standard sum Σ.)
There are powerful numerical summation methods that are neither regular nor linear, for instance nonlinear sequence transformations like Levin-type sequence transformations and Padé approximants, as well as the order-dependent mappings of perturbative series based on renormalization techniques.
Taking regularity, linearity and stability as axioms, it is possible to sum many divergent series by elementary algebraic manipulations.
However, when r is a real number larger than 1, the partial sums increase without bound, and averaging methods assign a limit of infinity.
Abelian means are regular and linear, but not stable and not always consistent between different choices of λ.
Then the limit of f(x) as x approaches 0 through positive reals is the limit of the power series for f(z) as z approaches 1 from below through positive reals, and the Abel sum A(s) is defined as Abel summation is interesting in part because it is consistent with but more powerful than Cesàro summation: A(s) = Ck(s) whenever the latter is defined.
If g(z) is analytic in a disk around zero, and hence has a Maclaurin series G(z) with a positive radius of convergence, then L(G(z)) = g(z) in the Mittag-Leffler star.
Several summation methods involve taking the value of an analytic continuation of a function.
If Σanxn converges for small complex x and can be analytically continued along some path from x = 0 to the point x = 1, then the sum of the series can be defined to be the value at x = 1.
One of the first examples of potentially different sums for a divergent series, using analytic continuation, was given by Callet,[5] who observed that if
The operation of Euler summation can be repeated several times, and this is essentially equivalent to taking an analytic continuation of a power series to the point z = 1.
If s = 0 is an isolated singularity, the sum is defined by the constant term of the Laurent series expansion.
If the series (for positive values of the an) converges for large real s and can be analytically continued along the real line to s = −1, then its value at s = −1 is called the zeta regularized sum of the series a1 + a2 + ... Zeta function regularization is nonlinear.
In applications, the numbers ai are sometimes the eigenvalues of a self-adjoint operator A with compact resolvent, and f(s) is then the trace of A−s.
[6] If J(x) = Σpnxn is an integral function, then the J sum of the series a0 + ... is defined to be if this limit exists.
There is a variation of this method where the series for J has a finite radius of convergence r and diverges at x = r. In this case one defines the sum as above, except taking the limit as x tends to r rather than infinity.
(If the numbers μn increase too rapidly then they do not uniquely determine the measure μ.)
Therefore, the summations are of the form This allows the usage of standard formulas for finite series such as arithmetic progressions in an infinite context.