Riemann Series Theorem - Generalization

Generalization

Given a converging series ∑ a_n of complex numbers, several cases can occur when considering the set of possible sums for all series ∑ a_σ (n) obtained by rearranging (permuting) the terms of that series:

• the series ∑ a_n may converge unconditionally; then, all rearranged series converge, and have the same sum: the set of sums of the rearranged series reduces to one point;

• the series ∑ a_n may fail to converge unconditionally; if S denotes the set of sums of those rearranged series that converge, then, either the set S is a line L in the complex plane C, of the form

or the set S is the whole complex plane C.

More generally, given a converging series of vectors in a finite dimensional real vector space E, the set of sums of converging rearranged series is an affine subspace of E.