# I Reversing the order of summation

1. Apr 15, 2016

### 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?

2. Apr 15, 2016

### BvU

No, since a+b = b+a

3. Apr 15, 2016

### henry wang

Thank you.

4. Apr 15, 2016

### 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

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.

5. Apr 16, 2016

### henry wang

Thank you.

6. Apr 16, 2016

### MAGNIBORO

look this video

7. Apr 16, 2016

### BvU

In post #1 there is.