generalsums

Generalsums is a term used in some expository contexts to denote a family of summation operators that generalize the ordinary sum by allowing variable weighting, transformation, or regularization of a sequence. The general setting associates a sequence {a_n} with a weight or transformation function w: N -> C, and defines the general sum S_w(a) as either the convergent sum sum_{n=1}^∞ w(n) a_n or, when convergence fails, a generalized value assigned by a summation method.

The simplest case is w(n) = 1 for all n, yielding the ordinary sum. Other choices include weightings w(n) = n^p, leading to weighted sums, or using a parameter to define a limiting process such as w(n) depending on a regulator. Generalized summation methods such as Cesàro, Abel, or Borel can be regarded as particular instances or compatible extensions within the generalsums framework, where a limit is taken after applying a regularization.

Convergence properties depend on the pair (a_n, w). Linearity holds: S_w(α a + β b) = α S_w(a) + β S_w(b) whenever the respective sums exist. The approach is useful for assigning finite values to divergent series or for studying asymptotic behavior by smoothing or weighting terms.

Historically, general summation concepts emerged in analysis to handle divergent series and to unify various techniques. In practice, generalsums appear in mathematical research, theoretical physics, and numerical methods as a way to model and manipulate series with flexible weighting schemes.