Rearranging series

group structures don't allow for infinitely applied operators

it's a purely sequential property

>>8526979
>If we look at R as algebraic structure, what is the main property that gives us this theorem?
The least upper bound property, and how any bounded closed set is compact. These properties form the basis for the proof that R is complete.

>>8527439
but that's a topological property,
>>8526986

this anon is right

>>8526979
If you are really interested in this shit you should look up this problem in Banach spaces, there are some weird shit there when it comes to rearranging series.

>>8527445
Im reading Kadets about that sort of problem. you think it's a good book?

And what property of metric function induces completeness?

>>8526979
How do you prove that
[math]\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{n} = \sum_{n=1}^{\infty} \frac 1n [/math]
?
Because
[math]\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^n}{n} = -\frac 11 + \frac 12 - \frac 13 + \frac 14 - \cdots[/math]
doesn't seem equal to me to
[math]\displaystyle \sum_{n=1}^{\infty} \frac 1n = \frac 11 + \frac 12 + \frac 13 + \frac 14 + \cdots[/math]

>>8528252
are you not familiar with the absolute value sign, friend?

>>8528252
amm it's a stupid bait but honestly, I didn't even look at the picture, it was first thing to pop up.

Im majoring in physics, should I know the answer to OPs question? Cause I don't

>>8528307
OP here. It's nothing too important. I was only curious how does summation operator behave algebraically on infinite sums and it turns out that it is not defined.

Apart from that, i started working with Banach spaces and it see that many of terms that I came across in my analysis course coincide but make completely different notions in general. (for example unconditional convergence and absolute convergence)