Examples of series that...

[elis_oc]

I am studying the sets \(\displaystyle l^2\) ("little ell-two", being the square-summable series) and \(\displaystyle l^1\) ("little ell-one", being the summable series). \(\displaystyle l^1\) is into \(\displaystyle l^2\), and \(\displaystyle C\) (the group of all convergent series) intersects \(\displaystyle l^2\) including the part of it that contains \(\displaystyle l^1\). Given this Venn diagram, I am trying to find an example of a series for all the possibilities that there are.

The three possible cases to consider are:
1) A series which is convergent, summable, and square-summable.
2) A series which is convergent and square-summable, but not summable.
3) A series which is divergent and not summable, but square-summable.

My teacher said the harmonic series (being the sum, from n=1 to infinity, of 1/n) is an example of number ( 3). Can anyone help me finding examples for (1) and (2)? Especially for number (2), since I think I have already found an example for number (1).

Thanks everyone!

 

[pka]

I am studying the sets l^2 (squared-sumable series) and l^1 (sumable-series). l^1 is into l^2 and C (the group of all convergent series) intersects l^2 including the part of it that contains l^1. Given this Venn diagram, I am trying to find an example of a serie for all the possibilities that there are.

The three possible cases to consider are:
1) A serie which is convergent, sumable and squared sumable.
2) A serie which is convergent Not sumable but it is squared sumable.
3)A serie which is divergent Not sumable but it is squared sumable.

My teacher said the harmonic succession (sum from n=1 to infinity of 1/n) is an example of number 3). Can anyone help me finding examples for 1) and 2)??.
Specially for number 2) since I think I have already found an example for number 1).

This is a very advanced question. Maybe it should be moved?
It is poorly posted. Are we to assume that the question is about Cearo summable series?
If so, what does squared-sumable serie meam? Abel sums?

 

[elis_oc]

This is a very advanced question. Maybe it should be moved?
It is poorly posted. Are we to assume that the question is about Cearo summable series?
If so, what does squared-sumable serie meam? Abel sums?

Yes, I also think it should be moved. Can someone tell me how is this done or where exactly should I move it? Sorry it is my first post in this web and still have not seen everything.

Ok, maybe squared-sumable serie wasn't the apropiate term, the objects belonging to l^2 are squared-sumable sequences since l^2 is a sequence space.

So a sequence of complex numbers z=(z_1,z_2,...) is squared-sumable if the series consisting in the sum from 1 to infinity of (the absolute value of z_n)^2 converges to an element of l^2. You can check it more precisely in here: http://en.wikipedia.org/wiki/Hilbert_space#Second_example:_sequence_spaces.
Hence a sequence is sumable if the series consisting in the sum from 1 to infinity of the absolute value of z_n converges to an element of l^1.

 

[lookagain]

square-summable

adj. (of a sequence) such that the series of the squares of its terms converges to a finite sum.

Source:

http://www.mathresources.com/products/mathresource/maa/squaresummable.html

 

[elis_oc]

Thank you all very much the serie you mentioned is what i was looking for!