Reversing the order of summation

[henry wang]

Is ∑f from a to b the same as ∑f from b to a?
In other words, does the order of summation matter?

 

[BvU]

No, since a+b = b+a

 

[henry wang]

No, since a+b = b+a

Thank you.

 

[zinq]

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

∑ c_j = K​

that does not converge absolutely:

∑ |c_j| = ∞,​

then there is an surprising theorem that suggests how important it is to be cautious:

Theorem: For such a summation as ∑ c_j, and any real number L, there is some rearrangement ∑' of the order of summation such that

∑' c_j = L.

 

[henry wang]

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

∑ c_j = K​

that does not converge absolutely:

∑ |c_j| = ∞,​

then there is an surprising theorem that suggests how important it is to be cautious:

Theorem: For such a summation as ∑ c_j, and any real number L, there is some rearrangement ∑' of the order of summation such that

∑' c_j = L.

Thank you.

 

[MAGNIBORO]

Thank you.

look this video

 

[BvU]

if there is only a finite number of terms

In post #1 there is.