# Reversing the order of summation

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

BvU
Homework Helper
No, since a+b = b+a

• henry wang
No, since a+b = b+a
Thank you.

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.

• 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

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.

Thank you.

look this video

• henry wang
BvU