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Every absolutely convergent complex series converges, and rearrangements preserve its sum
Statement
Every absolutely convergent complex series converges, and every rearrangement has the same sum. The conventions and prerequisite facts used below are recorded in Complex series, absolute convergence, complex power series, and radius of convergence, An absolutely convergent series in converges, and every rearrangement converges to the same sum.
Facts & Assumptions
Given: A complex series with convergent modulus series.
Proof
Regard its terms as vectors in ; the Euclidean norm is the complex modulus.
Apply the absolute-convergence and rearrangement theorem for finite-dimensional vector series.