The discussion centers around the question of whether the order of summation matters in mathematical series, specifically comparing the summation from a to b versus from b to a. Participants explore implications for both finite and infinite series, including conditions under which the order may or may not affect the result.
Participants express differing views on the impact of summation order, with some agreeing that it does not matter under certain conditions, while others highlight potential complications with infinite series. The discussion remains unresolved regarding the implications for infinite summations.
The discussion includes assumptions about the nature of the series being considered, particularly regarding convergence and the signs of terms. There are unresolved mathematical implications related to the rearrangement of infinite series.
Thank you.BvU said:No, since a+b = b+a
Thank you.zinq said:I'll add that, if there is only a finite number of terms, or if all but finitely many nonzero terms are of the same sign, then any order of summation gives the same result.
But (and I hope this is not too much information):
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For any convergent infinite summation
∑ cj = K
that does not converge absolutely:
∑ |cj| = ∞,
then there is an surprising theorem that suggests how important it is to be cautious:
Theorem: For such a summation as ∑ cj, and any real number L, there is some rearrangement ∑' of the order of summation such that
∑' cj = L.
henry wang said:Thank you.
In post #1 there is.zinq said:if there is only a finite number of terms