Vol. 73, No. 1, 1977

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The convergence-preserving rearrangements of real infinite series

Gerald Stoller

Vol. 73 (1977), No. 1, 227–231

Lel 𝒞 be the set of all permutations of the natural numbers that carry convergent real infinite series into convergent real infinite series. A strictly algebraic necessary and sufficient condition which determines 𝒞 is given. 𝒞 is seen to be a monoid but not a group. The maximum subgroup of 𝒞 is shown not to be normal in 𝒞.

A related set of permutations are those that preserve the sum of a convergent real infinite series when they carry that series to a convergent real series. This set of permutations is not a monoid. By exhibiting three different sufficient conditions for a permutation to belong to this set, we see that necessary and sufficient conditions determining this set will be difficult to ascertain.

Mathematical Subject Classification 2000
Primary: 40A05
Received: 1 June 1976
Revised: 5 January 1977
Published: 1 November 1977
Gerald Stoller