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Snaw

 Mathematics

Associated with Math.SE; for both general discussion & math qu...
3
Snaw
May 16, 2022 09:32
hmm, actually they might
Snaw
May 16, 2022 09:28
these are engineering students, I don't think they will learn about banach spaces
Snaw
May 16, 2022 09:28
it is indeed an odd omission. I mean they haven't even discussed Cauchy sequences of real numbers. They might have mentioned it, but they glossed over the concept and haven't for instance seen a proof that over the reals a sequence is convergent if and only if it is Cauchy.
Snaw
May 16, 2022 09:17
I'm gonna go think about how I can get rid of Cauchy's criterion for uniform convergence in this proof. My students didn't learn the basic Cauchy's criterion for convergence of sequences and series in the preceding course and so I'm trying to avoid anything that resembles it everywhere I can. I probably won't show them this result about uniform convergence anyway, but just in case someone asks about it.
Snaw
May 16, 2022 09:13
you should probably have something to eat then
Snaw
May 16, 2022 09:06
nowhere dense? :)
Snaw
May 16, 2022 09:05
I actually found this argument (in a slightly different form) in my lecture notes that discuss Abel's theorem. I think it can be changed a little to work without Cauchy's criterion, I have to think about that
Snaw
May 16, 2022 09:03
I should have been more explicit, the expression $|\sum_{k=n}^m a_k a^k|$ is also less than any given $\epsilon>0$ for any $n>N$ and $m$.
Snaw
May 16, 2022 09:02
no, why $0$? the sum is from $k=n$ to $m$, not from $1$ to $\infty$
Snaw
May 16, 2022 08:59
I probably shouldn't use this bad notation $a_k a^k$ :)
Snaw
May 16, 2022 08:58
@CalvinKhor assume by contradiction there is uniform convergence in say $(0,a)$. so by Cauchy's criterion for uniform convergence there's some $N$ such that for all $n>N$ and $m\in\mathbb{N}$ the absolute value of $\sum_{k=n}^m a_k x^k$ is less than any given $\epsilon$. Let $x\to a^-$ (this is a finite sum) so that the absolute value of $\sum_{k=n}^m a_k a^k$ is $\leq\epsilon$ so that $\sum_{k=1}^{\infty} a_k a^k$ converges contradicting the assumption that there is no convergence in the endpoint
2
Snaw
May 16, 2022 08:51
@Koro yes of course
Snaw
May 16, 2022 08:51
@CalvinKhor right but I meant to ask about when the convergence is strictly in $(-a,a)$ i.e. there is no convergence in the endpoints. Abel says if there is convergence in the endpoints, say in $-a$, then there will be uniform convergence in $[-a, b]$ for any $b<a$
Snaw
May 16, 2022 08:48
I think I have a proof that it can't converge uniformly in $(-a,a-\epsilon)$ for any $\epsilon>0$
Snaw
May 16, 2022 08:47
@CalvinKhor it's okay, thanks :) It was a silly question like I suspected.
Snaw
May 16, 2022 08:46
[[regarding my question above (I can't seem to reply to my own message)- never mind, I found the answer. Indeed it can't converge uniformly in $(-a,a)$.]]
Snaw
May 16, 2022 08:25
probably a silly question, but.. say a power series has interval of convergence $(-a,a)$ (i.e. it converges in $(-a,a)$ and nowhere outside this set). Does this necessarily mean the power series does not convergence uniformly on $(-a,a)$?
Snaw
Apr 4, 2022 19:05
@Koro I'm guessing the intention is that you should imagine that there is a separator between $1$ and $j$? So for $j=10$ you'd get $x_{1,10}$ or something like that. Like when indexing a matrix using $a_{ij}$ where $i$ is row and $j$ is a column, if you substituted $i=10$,$j=11$ you'd use some separator between them.
Snaw
Apr 4, 2022 18:43
@TedShifrin Oh, wonderful! Thank you!
Snaw
Apr 4, 2022 18:36
(Above I should have said $g(x)=(x-x_0)^{n+1}$.)
Snaw
Apr 4, 2022 18:35
I'm not looking for very deep motivation either. Is there a way I could play around with simple Taylor series and find this guess by myself after encountering enough examples?
Snaw
Apr 4, 2022 18:34
This isolates part of the problem, but it would still not be clear to me why I should use $g(x)=(x-x_0)^n$ or why it should be obvious to me that this result should be applied in order to find the remainder of the Taylor series.
Snaw
Apr 4, 2022 18:33
Yeah, I like the proof using Cauchy's MVT. It's very straightforward. My problem is how one would have guessed this formula in the first place. You mention generalizing the mean value theorem. I'm guessing you mean something like that if at $x_0$ we have that $f$ is $n+1$ times differentiable with its first $n$ derivatives vanishing then for every $x\ne x_0$ there is some $c$ between $x$ and $x_0$ such that $\frac{f(x)}{g(x)}=\frac{f^{(n+1)}(c)}{g^{(n+1)}(c)}$.
Snaw
Apr 4, 2022 18:27
I meant $R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)^{n+1}$ of course.
Snaw
Apr 4, 2022 18:27
Yeah, I was just about to fix it but I seem to not be able to edit the message.
Snaw
Apr 4, 2022 18:25
https://math.stackexchange.com/questions/3268002/intuitive-understanding-of-taylors-inequality-lagranges-remainder
But it didn't get any answers. There is one helpful comment but I don't find it too satisfying.
Snaw
Apr 4, 2022 18:24
How can I motivate Lagrange's remainder for Taylor series? Once one has the guess that $R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-c)^{n+1}$ for some $c$ between $x$ and $x_0$ the proof is very simple, just applying the mean value theorem again and again. But I wouldn't be able to start the proof if I hadn't first guessed that this is the form it should take. I thought of asking this question but it seems it would be a duplicate as it has already been asked before:
 

 CURED

For feedback/discussion/requests of Close/Undelete/Reopen/Edit...
2
Snaw
Apr 24, 2022 16:56
@OliverDíaz (I do not understand the somewhat aggressive or at least accusatory tone. I posted this question here myself a few days ago asking for criticism regarding whether it is a legitimate one. This conversation could have gone much differently.)
Snaw
Apr 24, 2022 16:56
@OliverDíaz When answering the question I believed it is an acceptable question here. The only thing to discuss is whether the question is acceptable or not. As you mention, I am active here, and so I try not to answer low quality questions. I was under the honest impression that this is not considered a low quality question. If I'm wrong I will re-adjust. But once again I do not see how your criticism on the quality of my answer have to do with anything.
Snaw
Apr 24, 2022 16:55
@OliverDíaz The heart of the problem is whether the question itself is legitimate or not. If the question is legitimate it should be answered rather than left unanswered. I see no reason for your attacks on the quality of my answer as it obviously answers the question, and I do not see the point of those attacks in your deleted comments blaming me for only caring about reputation and so on. This is uncalled for.
Snaw
Apr 21, 2022 02:38
@XanderHenderson Thanks for the information. While I'd still believe some sort of suspension or warning could prevent a borderline offensive trolling act from turning into something more serious next time, I do get your point about the situation being handled automatically and that moderators should be the "exception handlers". I guess it's also a sort of a visceral response on my part, expecting those who meant harm to be castigated.
Snaw
Apr 21, 2022 01:20
That user does have a long track of participation in SE, even if not in MSE, so their post seems weird. Maybe they had one puff too many on "420 day"..
Snaw
Apr 21, 2022 01:18
@XanderHenderson It wasn't just about getting the offensive post deleted quicker (which I assumed would happen sooner or later). Mainly I thought that if the post got quietly deleted by the community then the user itself would get away with the offensive act and then repeat the trolling again. I figured if I flag it for what it is then moderators could take action not only against the question but also against the "asker". (By the way, it was posted on what's called "420 day". Doubt that's incidental.)
Snaw
Apr 20, 2022 16:36
@XanderHenderson By the way, I see my flag was rejected. The reason I didn't mark it "rude or abusive" is because it didn't contain any profanity per se but only made these silly suggestive remarks.
Snaw
Apr 20, 2022 16:33
@XanderHenderson That's probably true, but consider that OP went out of their way to include suggestive remarks 3 times (including in their answer), so that it might make someone go ahead and search what the whole thing is about.
Snaw
Apr 20, 2022 16:23
This post with its sexually suggestive remark should be deleted
Snaw
Apr 18, 2022 13:20
@postmortes Thanks! I didn't know there was a relationship between the score of the answers and the ability to delete the question. Personally it doesn't seem like a PSQ to me as the attempt at the solution was given and so it seems like a standard solution-verification type question. I understand it could be a matter of personal taste. (I agree my answer doesn't add too much to Kavi's comment, but I thought it's better to have it answered rather than be left without an answer.)
Snaw
Apr 18, 2022 08:13
Since I posted my answer asking for feedback whether there is something wrong with it, the question it answers received 3 close votes (and I got one more downvote). Being that this is CURED I think it's just the right place to discuss this - that's the reason I posted that here - so I can hear your opinions. Isn't it a standard solution-verification question of which we see tens of them posted (and keep open) daily? What makes this different?
Snaw
Apr 17, 2022 18:22
Do you see anything wrong with my answer here? It was just downvoted without a comment, and assuming this wasn't out of spite I'm wondering if there is either an error I'm missing or else if someone thought this question was too simple to be answered. However, it seems like a standard solution-verification question which seem to be accepted here. Also the question itself wasn't downvoted or voted to be closed, so I'm unsure what the intent was.
Snaw
Apr 17, 2022 06:59
Close: Duplicate (by user with many low quality contributions).
Snaw
Apr 14, 2022 23:29
Duplicate
Snaw
Apr 14, 2022 13:20
A bunch of PSQ's: Close 1, Close 2, Close 3, Close 4, Close 5, Close 6, Close 7
Snaw
Mar 29, 2022 21:31
Another PSQ by the same user (C/D).
Snaw
Mar 29, 2022 18:53
@Peter This one is especially annoying. It is a PSQ which turns into a rant about how PSQs are dealt with, and it concludes with a blatant lie ("the problem just came into my mind", regarding $1992!+a$ being a perfect power. Yeah, right.)
Snaw
Mar 25, 2022 13:01
@Peter Why not? Seems fine to me
 

 Math Mods' Office

For informal chat with the site moderators about moderation, s...
2
Snaw
Apr 21, 2022 02:40
@ParamanandSingh Noted. And I'll cast close votes when flagging for EoQS. Thanks!
Snaw
Apr 20, 2022 17:58
@XanderHenderson OK. Thanks!
Snaw
Apr 20, 2022 17:53
Should I continue flagging low quality contributions by this user or will it not be helpful at this point?
Snaw
Apr 20, 2022 17:53
@XanderHenderson I understand.
Snaw
Apr 20, 2022 16:43
@XanderHenderson A few days ago I reported some EoQS violations in this chatroom. I see my message was deleted (assuming I didn't post in the wrong chatroom..)
Was I not supposed to post this message here? I flagged a bunch of the posts by the same user. I got the message that there is no reason to act because recent answers by this OP are fine. But looking at answers from just the last couple of days shows them answering blatant PSQ's that don't even make an attempt at hiding their PSQ status. I can flag all those low quality contributions by them, if that is helpful, but I won't go throug…