Reversing the order of summation

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  • Thread starter henry wang
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  • #1
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Is ∑f from a to b the same as ∑f from b to a?
In other words, does the order of summation matter?
 

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  • #2
BvU
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No, since a+b = b+a
 
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  • #4
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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):
-----------------------------------------------------------

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.
 
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  • #5
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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):
-----------------------------------------------------------

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.
Thank you.
 
  • #7
BvU
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if there is only a finite number of terms
In post #1 there is.
 

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