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.
