Timeline for Two Equal Series?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 11, 2010 at 4:10 | answer | added | fedja | timeline score: 6 | |
| Dec 10, 2010 at 21:37 | answer | added | Tony Huynh | timeline score: 1 | |
| Dec 10, 2010 at 19:34 | answer | added | Pietro Majer | timeline score: 3 | |
| Dec 10, 2010 at 16:51 | comment | added | backstoreality | @t3suji: Not at all, it was of my own interesting @Andres and Sergi: Yes, I would have been better specifying the summation $\sum_{n}\frac{1}{a_n^s}$ and $\sum_{n}\frac{1}{b_n^s}$ and giving the condition that all $a_n,b_n$ must lie outside the unit disk in the complex plane so as to ensure that the summations are finite | |
| Dec 10, 2010 at 16:01 | comment | added | Sergei Ivanov | I don't know about prior art, but here is a simple proof (assuming they are positive as per Andres Caicedo's comment). Rearrange in descending order, find the first mismatch and remove the preceding terms. Now $a_1>b_1\ge b_i$ for all $i$. Multiply by a constant so that $a_1=1$. Now the first sum is at least 1 but the second goes to 0 as $s\to\infty$. | |
| Dec 10, 2010 at 15:56 | answer | added | Fedor Petrov | timeline score: 8 | |
| Dec 10, 2010 at 15:46 | comment | added | Andrés E. Caicedo | @Joe : I think you need some additional assumption. Are the $a_n,b_n$ also supposed to be positive? Otherwise, you can start with, say, $a_n=1/n^2$, and add infinitely many zeros in any order you want, to form the $b_n$. | |
| Dec 10, 2010 at 15:05 | comment | added | t3suji | Out of curiosity: is the question by chance inspired by one of the problems on the latest Putnam? | |
| Dec 10, 2010 at 14:31 | history | asked | backstoreality | CC BY-SA 2.5 |