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!
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!
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