Timeline for Associativity of infinite products
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 25 at 9:43 | comment | added | Hilbert Jr. | I'm saying that if $\prod_nu_n$ is your product, then to show it converges absolutely you have to show $\prod_n(1+|u_n-1|)$ converges, right? Since if $u_n=1+w_n$, then $w_n=u_n-1$. Well in our case the individual factors are $u_n=\prod_{j\in J_n}(1+w_j)$, so you'd have to show $\prod_n(1+|(\prod_{j\in J_n}(1+w_j))-1|)$ converges to show $\prod_n\prod_{j\in J_n}(1+w_j)$ converges absolutely, no? Or do I have it wrong somehow? | |
| Jun 25 at 8:20 | comment | added | Joshua Tilley | I don't agree, can you explain why? That is a different product. | |
| Jun 25 at 8:11 | comment | added | Hilbert Jr. | I believe I was overhasty in accepting your answer: by definition $\prod_n(1+w_n)$ is absolutely convergent if $\prod_n(1+|w_n|)$ convergers, right? Well in our case the product is $\prod_n(\prod_{j\in J_n}(1+w_j))$, so you'd have to show that $\prod_n(1+|(\prod_{j\in J_n}(1+w_n))-1|)$ converges, no? | |
| Jun 24 at 23:10 | history | edited | Joshua Tilley | CC BY-SA 4.0 |
deleted 13 characters in body
|
| Jun 24 at 16:47 | history | edited | Joshua Tilley | CC BY-SA 4.0 |
added 811 characters in body
|
| Jun 24 at 15:17 | comment | added | Joshua Tilley | You're welcome! The rough strategy for infinite products I would suggest is to take logs so you are always thinking in terms of sums only. Then you need to estimate $\log(1+w_i)$ which I have addressed in my answer. | |
| Jun 24 at 8:08 | comment | added | Hilbert Jr. | Thank you! I was so fixated on proving convergence directly from the series criterion, I completely missed this. | |
| Jun 24 at 8:07 | history | bounty ended | Hilbert Jr. | ||
| Jun 24 at 8:07 | vote | accept | Hilbert Jr. | ||
| Jun 27 at 9:32 | |||||
| Jun 23 at 13:31 | history | edited | Joshua Tilley | CC BY-SA 4.0 |
added 600 characters in body
|
| Jun 23 at 11:37 | comment | added | Hilbert Jr. | I am aware of this criterion for convergence, I mention it in the question, and it's easily applied to show that each of the products $\prod_{j\in J_n}z_j$ converges absolutely. My problem, as outlined in the question, is showing that the entire product $\prod_{n=1}^\infty\prod_{j\in J_n}z_j$ converges. Its absolute convergence is, by the theorem you mention, equivalent to the convergence of the series $\sum_{n=1}^\infty|(\prod_{j\in J_n}z_j)-1|$, no? My problem is I don't see why this series should converge. | |
| Jun 23 at 10:51 | history | edited | Joshua Tilley | CC BY-SA 4.0 |
added 6 characters in body
|
| Jun 23 at 10:45 | history | answered | Joshua Tilley | CC BY-SA 4.0 |