Description: Define the sum of a
series with an index set of integers .
is normally a free variable in , i.e. can be thought of as
   . This definition is the result of a collection of
discussions over the most general definition for a sum that does not
need the index set to have a specified ordering. This definition is in
two parts, one for finite sums and one for subsets of the upper
integers. When summing over a subset of the upper integers, we extend
the index set to the upper integers by adding zero outside the domain,
and then sum the set in order, setting the result to the limit of the
partial sums, if it exists. This means that conditionally convergent
sums can be evaluated meaningfully. For finite sums, we are explicitly
order-independent, by picking any bijection to a 1-based finite sequence
and summing in the induced order. In both cases we have an
expression so that we only need to be defined where
.
In the infinite case, we also require that the indexing set be a
decidable subset of an upperset of integers (that is, membership of
integers in it is decidable). These two methods of summation produce
the same result on their common region of definition (i.e. finite sets
of integers). Examples:      means
, and        means
1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 11130). (Contributed by NM,
11-Dec-2005.) (Revised by Jim Kingdon,
21-May-2023.) |