summability

Summability is a branch of mathematical analysis that studies methods for assigning finite sums to series that do not converge in the usual sense. Given a series ∑ a_n with partial sums s_n = ∑_{k=0}^n a_k, a summability method provides a rule to assign a value S, which may differ from the ordinary sum, to the sequence of partial sums or to the series itself.

Regularity is a key concept: a summability method is regular if every ordinarily convergent series converges to the same sum under the method. Regular methods preserve ordinary convergence, while nonregular methods extend summation to divergent series.

Common summability methods include:

- Cesàro summation, which looks at the averages σ_n = (s_0 + s_1 + … + s_n)/(n+1) and defines the sum as lim σ_n when this limit exists.

- Abel summation, which assigns the sum as the limit, if it exists, of ∑_{n=0}^∞ a_n x^n as x approaches 1 from below.

- Borel summation, which uses a Borel transform ∑ a_n t^n / n! and a subsequent Laplace-type integral to obtain a sum, when possible.

Other methods, such as Dirichlet, Euler, and Riesz summation, extend these ideas in various ways. Summability theory finds applications in Fourier analysis, asymptotic expansions, and number theory, and it interacts with convergence through Tauberian theorems that link summability properties to ordinary convergence under additional hypotheses.

A classic illustration is the series 1 − 1 + 1 − 1 + …, which is divergent in the traditional sense but is Cesàro- and Abel-summable to 1/2.