Conditional vs unconditional convergence

[johnqwertyful]

NOT talking about nonabsolute vs absolute convergence. I'm talking about conditional convergence. In my analysis text, this was a bit that was covered as enrichment and it straight up blew my mind. I don't get it. How can you simply rearrange terms and come up with a separate sum? They showed a few examples in the book, but it still blew my mind.

I don't really have a question. I just find this idea awesome. Also Reimann Series Theorem blew my mind.

Does anyone know much about it? The book I had, although clear, wasn't as deep as I would have liked.

 

[chiro]

Hey johnqwertyful.

It's basically what happens when you deal with infinity and this kind of thing is problematic in many areas of mathematics that analyze it.

In the conditional case, there are infinitely many terms and infinitely many ways to evaluate them but the crux is that you can't fall into the finitistic way of thinking.

The finitistic way of thinking is that you can divide an infinite series into finitely many terms whether its through sums, products or combinations thereof.

This is why when people try to decompose some series and evaluate it in a specific finitistic way (like say collecting terms and representing them as n objects) then you fall into trouble.

It's not just way arithmetic or series though: the same thing happens with trying to make sense of infinity wherever it's applied whether that include how you can decompose for example, an infinite linear space (like a Hilbert-Space): if you are not careful you can fall into the same trap.

It might help you to keep in mind of getting caught in the same trap of taking something infinite and trying to make it finite: in some cases you can, but in general you can't and if you're not careful you will run into the same mistakes yourself.

 

[johnqwertyful]

Interesting man. Thanks. I guess the main thing I've learned from analysis is that I really don't know squat about things I thought I knew. Infinity, convergence, limits, real numbers.

I'm super excited for my analysis class to start. I've already worked through a few chapters in my book, and now have been skipping around. It's a fantastic book. Crystal clear, challenging but doable problems, no problem is trivial.

http://books.google.com/books?id=Wlb-o7HBh2YC&printsec=frontcover#v=onepage&q&f=false