Riemann series theorem

The theorem states that the terms of a conditionally convergent real series can be rearranged to converge to any real number or to diverge.

Conditional convergence: A series converges, but the series of absolute values of its terms diverges. The theorem applies precisely when convergence depends on the original ordering.
Infinite series: An infinite series is the limit, when it exists, of the partial sums of a sequence of terms. The theorem concerns how permuting a series’ terms changes its partial-sum limit.
Bernhard Riemann: Bernhard Riemann was a German mathematician whose work shaped analysis, geometry, and number theory. His name is attached to the theorem on rearranging conditionally convergent series.
Commutativity: Commutativity is the property that changing the order of an operation’s inputs leaves its result unchanged. Finite addition is commutative, but conditionally convergent infinite sums need not preserve that behavior.
Absolute convergence: A series converges absolutely when the series of absolute values of its terms converges. Absolute convergence prevents rearrangements from changing a series’ sum.
Convergence of a series: A series converges when its sequence of partial sums has a finite limit. The original series must converge before its rearranged limits can be compared.
Niels Henrik Abel: Niels Henrik Abel was a Norwegian mathematician known for foundational work in analysis and algebra. Abel’s results on series convergence form part of the nineteenth-century background to rearrangement theory.
Tonelli's theorem: Tonelli's theorem permits exchanging sums or integrals of nonnegative functions, even when the value is infinite. Nonnegativity restores safe order changes in a setting closely related to rearranged sums.
Positive and negative subseries: The positive and negative subseries are formed by separating a real series’ positive and negative terms. For conditional convergence, both separated subseries diverge in magnitude, enabling controlled overshoots.
Series of absolute values: The series of absolute values replaces each term of a real series by its magnitude. Its divergence distinguishes the theorem’s conditionally convergent series from stable ones.

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