Comparing Sets of Convergent Sequences and Series

[Char. Limit]

So I had this question in PF chat, but I decided this would be a better place for it.

Say I have two sets, S and S'. S is the set of all convergent sequences. S' is the set of all convergent series...es.

Is S larger than S', and if so, how much larger?

 

[snipez90]

This question doesn't make much legitimate sense to me. Any convergent series is a limit of partial sums and hence S' is contained in S. But it doesn't seem that useful to ask how much larger S is than S' simply because many sequences are just not sequences of partial sums.

 

[chiro]

So I had this question in PF chat, but I decided this would be a better place for it.

Say I have two sets, S and S'. S is the set of all convergent sequences. S' is the set of all convergent series...es.

Is S larger than S', and if so, how much larger?

Have you done any work/research in l^2(R) spaces, L^2(R) spaces and hilbert spaces?

 

[Char. Limit]

Have you done any work/research in l^2(R) spaces, L^2(R) spaces and hilbert spaces?

No, but if you can tell me about them a bit I'll try my best to understand.

 

[chiro]

No, but if you can tell me about them a bit I'll try my best to understand.

l^2(R) spaces represent the set of sequences such that the norm of the sequence (think a vector with infinite dimensions) is finite.

So basically if you take every element in your sequence, square it, and all of them together, the result is finite. Since that implies a notion of convergence for some series, it sounds like you may find what you're looking for by looking properties of l^2(R).

Hilbert spaces basically have the idea that you can have sequences of sequences and while the sequence in one element converges, so does the norm as well. Hilbert spaces are actually more correctly spaces that are "complete inner product spaces", meaning that they have a valid inner product with completeness properties.

If you want to understand completeness you need to understand what a Cauchy sequence is. Its basically a way of defining convergence with your standard norm definition (think epsilon-delta where as you get closer to a particular value from your set, norm gets smaller) with the exception that you are using a different set of parameters to define convergence. I'm yet to learn latex so I'll direct you to this page:

http://en.wikipedia.org/wiki/Cauchy_sequence

At the moment I'm doing a course on wavelets so we do touch on a few areas of analysis, but nothing in too much depth. Based on what I've learned you're best bet is to look at l^2(R) and look at "complete" spaces, at least "complete normed spaces" (complete normed spaces are called Banach spaces, and complete inner product spaces are called Hilbert spaces. All inner product spaces are normed spaces, but not all normed spaces are inner product spaces). You'll probably get a lot of results that might help you, but again this is just an educated guess based on what I'm learning.

 

[micromass]

Why do all the interesting questions in chat get asked when I'm away?

To answer your question without using $$\ell^2$$ or the like: the sets have exactly thesame size. That is, there is a way to transform a convergent sequence into a series and vice versa.

Given a convergent series, $$\sum{u_n}$$, then we denote the partial sums $$s_n=\sum_{k=1}^n{u_k}$$. This partial sums form a sequence $$(s_n)_n$$ that converges.

The surprising thing is that the other way around also works. Given a sequence $$(a_n)_n$$, then we denote $$u_0=a_0$$ and $$u_n=a_n-a_{n-1}$$. Then $$\sum{u_n}$$ forms a series with partial sums the $$(a_n)_n$$.

Thus every convergent sequence determines uniquely a convergent series, and vice versa. So to study series, it is actually enough to study sequences...

 

[snipez90]

Ah thanks, I didn't think carefully through the other direction. The obvious direction follows from the definition and that alone explains why the study of series is based on the study of sequences.

 

[chiro]

Why do all the interesting questions in chat get asked when I'm away?

To answer your question without using $$\ell^2$$ or the like: the sets have exactly thesame size. That is, there is a way to transform a convergent sequence into a series and vice versa.

Given a convergent series, $$\sum{u_n}$$, then we denote the partial sums $$s_n=\sum_{k=1}^n{u_k}$$. This partial sums form a sequence $$(s_n)_n$$ that converges.

The surprising thing is that the other way around also works. Given a sequence $$(a_n)_n$$, then we denote $$u_0=a_0$$ and $$u_n=a_n-a_{n-1}$$. Then $$\sum{u_n}$$ forms a series with partial sums the $$(a_n)_n$$.

Thus every convergent sequence determines uniquely a convergent series, and vice versa. So to study series, it is actually enough to study sequences...

Very insightful. Thankyou :)