On convergent series - in the spirit of Abel and Dini
The following is something that I think and hope is true. I tried searching for it online but with little success. A reference, proof, or counterexample would be nice.
Suppose $a_i $ is a positive sequence and $sum_i a_i < infty.$ Then there exists a positive sequence $b_i$ s.t $sum_i b_i < infty$ and $sum_i frac{a_i}{b_i} < infty$.
sequences-and-series reference-request
asked Dec 1 '18 at 13:54
Better2BLucky
586
add a comment |
The following is something that I think and hope is true. I tried searching for it online but with little success. A reference, proof, or counterexample would be nice.
Suppose $a_i $ is a positive sequence and $sum_i a_i < infty.$ Then there exists a positive sequence $b_i$ s.t $sum_i b_i < infty$ and $sum_i frac{a_i}{b_i} < infty$.
sequences-and-series reference-request
asked Dec 1 '18 at 13:54
Better2BLucky
586
This is true, you can take $b_n=sqrt {sum_{i=n}^{infty}a_i}$. See this question or this one for more information and some relevant links.
– lulu
Dec 1 '18 at 14:27
@lulu I saw this post - I don't think it's neceassily true that $sum_i b_i < infty$ if we do that
– Better2BLucky
Dec 1 '18 at 14:31
Yes....I don't see how to show that. Indeed, it looks like it might not be hard to make a counterexample to that. Odd. Rubin's claim regarding that series isn't very interesting if the sum diverges (not a proof of anything, clearly)
– lulu
Dec 1 '18 at 14:50
add a comment |
The following is something that I think and hope is true. I tried searching for it online but with little success. A reference, proof, or counterexample would be nice.
Suppose $a_i $ is a positive sequence and $sum_i a_i < infty.$ Then there exists a positive sequence $b_i$ s.t $sum_i b_i < infty$ and $sum_i frac{a_i}{b_i} < infty$.
sequences-and-series reference-request
asked Dec 1 '18 at 13:54
Better2BLucky
586
The following is something that I think and hope is true. I tried searching for it online but with little success. A reference, proof, or counterexample would be nice.
Suppose $a_i $ is a positive sequence and $sum_i a_i < infty.$ Then there exists a positive sequence $b_i$ s.t $sum_i b_i < infty$ and $sum_i frac{a_i}{b_i} < infty$.
sequences-and-series reference-request
sequences-and-series reference-request
asked Dec 1 '18 at 13:54
Better2BLucky
586
asked Dec 1 '18 at 13:54
Better2BLucky
586
edited Dec 1 '18 at 14:01
asked Dec 1 '18 at 13:54
Better2BLucky
586
asked Dec 1 '18 at 13:54
Better2BLucky
586
asked Dec 1 '18 at 13:54
Better2BLucky
586
586
This is true, you can take $b_n=sqrt {sum_{i=n}^{infty}a_i}$. See this question or this one for more information and some relevant links.
– lulu
Dec 1 '18 at 14:27
@lulu I saw this post - I don't think it's neceassily true that $sum_i b_i < infty$ if we do that
– Better2BLucky
Dec 1 '18 at 14:31
Yes....I don't see how to show that. Indeed, it looks like it might not be hard to make a counterexample to that. Odd. Rubin's claim regarding that series isn't very interesting if the sum diverges (not a proof of anything, clearly)
– lulu
Dec 1 '18 at 14:50
add a comment |
This is true, you can take $b_n=sqrt {sum_{i=n}^{infty}a_i}$. See this question or this one for more information and some relevant links.
– lulu
Dec 1 '18 at 14:27
@lulu I saw this post - I don't think it's neceassily true that $sum_i b_i < infty$ if we do that
– Better2BLucky
Dec 1 '18 at 14:31
Yes....I don't see how to show that. Indeed, it looks like it might not be hard to make a counterexample to that. Odd. Rubin's claim regarding that series isn't very interesting if the sum diverges (not a proof of anything, clearly)
– lulu
Dec 1 '18 at 14:50
This is true, you can take $b_n=sqrt {sum_{i=n}^{infty}a_i}$. See this question or this one for more information and some relevant links.
– lulu
Dec 1 '18 at 14:27
This is true, you can take $b_n=sqrt {sum_{i=n}^{infty}a_i}$. See this question or this one for more information and some relevant links.
– lulu
Dec 1 '18 at 14:27
@lulu I saw this post - I don't think it's neceassily true that $sum_i b_i < infty$ if we do that
– Better2BLucky
Dec 1 '18 at 14:31
@lulu I saw this post - I don't think it's neceassily true that $sum_i b_i < infty$ if we do that
– Better2BLucky
Dec 1 '18 at 14:31
Yes....I don't see how to show that. Indeed, it looks like it might not be hard to make a counterexample to that. Odd. Rubin's claim regarding that series isn't very interesting if the sum diverges (not a proof of anything, clearly)
– lulu
Dec 1 '18 at 14:50
Yes....I don't see how to show that. Indeed, it looks like it might not be hard to make a counterexample to that. Odd. Rubin's claim regarding that series isn't very interesting if the sum diverges (not a proof of anything, clearly)
– lulu
Dec 1 '18 at 14:50
add a comment |
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This is true, you can take $b_n=sqrt {sum_{i=n}^{infty}a_i}$. See this question or this one for more information and some relevant links.
– lulu
Dec 1 '18 at 14:27
@lulu I saw this post - I don't think it's neceassily true that $sum_i b_i < infty$ if we do that
– Better2BLucky
Dec 1 '18 at 14:31
Yes....I don't see how to show that. Indeed, it looks like it might not be hard to make a counterexample to that. Odd. Rubin's claim regarding that series isn't very interesting if the sum diverges (not a proof of anything, clearly)
– lulu
Dec 1 '18 at 14:50