The Riemann Rearrangement Theorem states that if an infinite series of real numbers converges conditionally, meaning it converges but not absolutely, then its terms can be reordered to sum to any chosen real number, or to diverge, purely by rearranging the order of addition. Proved by Bernhard Riemann, it shows that the ordinary commutative law of addition fails for infinite sums that are not absolutely convergent.
Facts
StatementIf an infinite series of real numbers is conditionally convergent, meaning it converges but not absolutely, then its terms can be reordered into a permutation that converges to any chosen real number, or rearranged so that the new series diverges instead; a real series is therefore unconditionally convergent only when it converges absolutely. 1 Connections
Sources
1. Riemann Rearrangement Theorem (Wikipedia)
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In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges.
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