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SumOverSet

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Description

Summary

The value of a sum over a set, sum(f(x), x in S): the terms written out and added
where the members of S are known and there are finitely many of them, and the sum
left as written otherwise.

Remarks

A sum over a set counts each member once, which is what makes it well defined without an
order: a set has no first member, and addition does not care. So the members have to be
known to be distinct before they are added, and a listed set is read only where its
elements are numbers. sum(x, x in { a, b }) is a + b when a and
b differ and a when they are equal, and nothing here can tell which, so it is
left as written.
 The sum over the roots of a polynomial is the case the node exists for:
sum(f(w), w in { w : p(w) = 0 }). Where every root is rational, a root of a
quadratic or a root of a binomial, the solver writes them and they are added; every is checked, by counting the distinct roots it gave against the degree of the square-free
part of p, which is how many distinct roots p has. The roots of an
irreducible cubic or quartic are not written out in radicals, which read no more simply
than the sum. A root set the solver answers only in part is left as written, since a sum
that misses a term is a wrong answer. Evaluated to a number, the roots the solver cannot
write are found to the working precision, all of them or none, by
DurandKerner. And a sum over a set that is not finite -- an
interval, the integers -- is a series, which is not this node's to answer.
#1285

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