@Sawarnik $e$ ?

okay...

@ParthKohli A

No B.

This is a more-than-one-correct question.

But B is the best option I think.

@ParthKohli Am I right?

@ParthKohli definitely if something is greater than A then it is greater than B too.

so, A and B

B is one, what's the other?

A is the other

A of course.

Show solution.

Ok..

Just a min

How do you claim that $\alpha$ has no upper bound?

Actually, this is nothing but a simply varied version of the AM-GM inequality situation...

$ax^2+bx+c$ is always positive means $c>\frac{b^2}{4a}$. And do this with the others and add.

@Hawk Yes.

@Hawk But it gave you a not the better bound.

@Sawarnik meaning?

@Hawk Means you proved its greater than 1, but the correct answer would have got its greater than 1/4.

@ParthKohli Is my solution good?

@Sawarnik Why? If you get something greater than $\dfrac14$ how can you claim it is greater than 1?

@Sawarnik but you can definitely claim the other way round.

:14923219 How do you get that it's greater than 1?

@DanielFischer please see the problem I posted to you before.

@Hawk @ParthKohli Will you like some geometry?

@Sawarnik @ParthKohli If $0<x<1$ and $A_n=\dfrac{x}{1-x^2}+\dfrac{x^2}{1-x^4}+\ldots+\dfrac{x^{2^n}}{1-x^{2^{n+1}}}$ then $\lim\limits_{n\to \infty}A_n$ is ?

@Sawarnik no, seems like my brain is not working...was asking all stupid questions to Daniel Fischer and r9m and even failed to answer the questions given by you.

@Hawk It's tough!

@Sawarnik Its very easy...just think a little.

Clearly you do not expect the answer to come in numericals, but in $x$ because there is nothing said about $x$ but its range. So, think now!

@r9m how can I approach $\sum\limits_{n=1}^{\infty}\dfrac{1}{(2n-1)^2}$ similar to the solution of Basel Problem by expansion of $\sin$ series

how?

@Hawk Was me right?

@Sawarnik Yes, but what is your approach?

@Hawk Seperate the even terms and factor out 1/4, so your series becomes 2pi^2/9. Is it correct?

@Sawarnik I cannot understant much without latex...can you post it more properly.

First of all is my answer correct?

no, it is not

Oh :(

You can try this way...since this is infinite series, you can write it as $\sum \dfrac{1}{n^2}=\sum\dfrac{1}{(2n)^2}+\sum\dfrac{1}{(2n-1)^2}$ which gives $\dfrac{\pi^2}{6}-\dfrac{\pi^2}{24}=\dfrac{\pi^2}{8}$

This is what I did ... and made a silly mistake! Yes, you are right.

Okay...so approach was correct

@Sawarnik Try to think of infinite GP series

@Hawk I couldn't think much :(

Okay, a hint?

@Sawarnik and anything from Bartle?

@Hawk Bartle was too hard.

@Sawarnik Which one was hard, Bartle or the problem?

@Hawk Would it be okay to skip the first two chapters of all the formal stuff, and go to interesting things like sequences?

@Sawarnik Its okay, if you can understand them...but if you feel something you can come back to rectify your concepts.

@Sawarnik could you do the problem now?

I'll be back...but will take some time....

@Hawk Ok bye... and I yes, I can do it.

@Chris'ssis: are you trying to show the convergence of that sum with those integrals?

Ah, I have to walk the dog. Be back in a bit

@r9m thanks

Hello there earthlings.

@robjohn Yeah. The left thing is the error term.

Does the integral $$ \int_0^\infty \frac{\mathrm{d}x}{1+x^n} $$ has a particular name?

It is very famous, and similar integrals have different names to be able to distinguish / refer to them easier.

@N3buchadnezzar You may call it like your name, N3buchadnezzar's integral.

We have the fresnell integrals, the first and second euler integral, the frullani integrals, the dirichlet integral, but this one is nameless.

Poor fellow

@Chris'ssis So you still think the sum converges?

@robjohn I don't say that, but I try to see why I was wrong.

@N3buchadnezzar this answer says that that integral is $\frac\pi{n}\csc\left(\frac\pi{n}\right)$, but I don't know a common name for it.

@robjohn I know its value, i do not know it's name ;)

@Chris'ssis you are trying to make the sum into a Riemann sum?

@robjohn I think of all variants. Do you think it works that way?

@Chris'ssis It would be a Riemann Sum for $\int_1^\infty\frac{\sin(x)}{x}\mathrm{d}x$ and that does converge, so the usual criterion for applying Riemann sums to improper integrals does not hold.

@robjohn I just found this question that might help. math.stackexchange.com/questions/630890/…

@robjohn Someone says in a comment that $$\sum _{n=2}^{\infty } \frac{\sin (\log (n))}{n \log (n)}=\Im\left(\int_1^{\infty } (-1+\zeta (x-i)) \, dx\right)\approx 1.14964647671184932067946796534639573228838143131427$$ This is similar to my series.

I was deleted from wikipedia. :( buhuu

@Chris'ssis I have a series for you.

@Chris'ssis yes, that is the integral that I show converges if the original sum converges.

sin vs cos is the same

=0.4790882572765523426...

Do you know any other form for it?

@MatsGranvik OK. I'll look of that when I finish with my series. Does it have a closed form?

It converges to that number above, that is all I know.

ok

@Chris'ssis I believe it does, I will work on it now.

@Chris'ssis oops, I was looking at it slightly wrong. I now don't know if it does.

@MatsGranvik What is the part of minus inside the $\cos(x)$?

@MatsGranvik Why?

@MatsGranvik The value of $k$?

k=2

@MatsGranvik Ah, I didn't see it above.

Sigh, where does the minus sign come from here ? chat.stackexchange.com/rooms/36/mathematics

Second line $\mathrm{Res}(z_k) = -\frac{1}{n}e^{i\pi(2k+1)/n}$, $0 \leq k \leq \frac{n}{2}-1$

@Chris'ssis It could be there is no closed form for it.

Other than: $$\sum_{n=1}^\infty(-1)^{n+1}\frac{\cos\left(\frac{4\pi }n\right)}{n}$$ as pointed out by Sami Ben Romdhane.

@MatsGranvik Does it converge? I would think not

Hi, I got stuck on something probably very trivial: sesquilinear forms are linear in either first or second argument. Why do we have to change the definition of the kernel (I heard its called left and right radical) if we change the argument of linearity? I thought it shouldn't be a subspace then, but can't prove it, since it seems to me that antilinearity ensures that both radicals are subspaces!

@MatsGranvik The sum can be accelerated quite a bit to make it easier to compute, but I am doubting a closed form.

@robjohn I have to say your way is the best way to my question. I don't imagine now how I might beat this way.

Oh, I think I know now. If not properly defined, the kernel might be empty -> not a subspace.

Rapture Palooza looks extremely good, watching it now on nf

@mirgee it's a subspace without identity

@EnjoysMath What is identity

rapture palooza

@Chris'ssis I just checked and since $\int_1^\infty|f'(x)|\,\mathrm{d}x$ does not converge, when $f(x)=\frac{\sin(x)}{x}$, using Riemann Sums does not apply to the infinite sum.

$0$ in case of a vs usually

@EnjoysMath Sorry :D

@EnjoysMath get it

@Chris'ssis that is why my initial attempt (when talking to Pedro) using Riemann sums showed convergence, but it did not apply.

@robjohn Look at this question math.stackexchange.com/questions/190966/…

@robjohn In certain conditions we may use it I think. (or I wrongy understand your point)

What's the question?

@EnjoysMath Test for convergence $$\sum_{k=1}^\infty\frac{\sin(H_k)}{kH_k}$$ (it's a question I created these days)

@Chris'ssis well the condition I gave above holds for $f(x)=\frac1{1+x^2}$

@robjohn Ah, right.

What's $H_k$?

@EnjoysMath $k$th harmonic number

I just answered another lhf.

(Doing it Jasper-style.)

lhf?

Low Hanging Fruit (relatively easy question)

@EnjoysMath How do you know it lacks the identity? Because h(0,x) = 0 and h(x,0) = 0, no matter the argument of linearity

Oh, it already had an answer.

@EnjoysMath $\displaystyle H_n=\sum_{k=1}^n\frac1k$

tanx

OK, well, I did it the inequalities-way.

@EnjoysMath Please, any hint would be very appreciated

@Chris'ssis maybe take the taylor series for $\sin(H_k)$ and show some relation with the other $\sin(H_{k+a})$'s ??

@mirgee what's ur quest?

@EnjoysMath @robjohn already gave a proof above

oh

@EnjoysMath Why left or right radical will lack the identity element (as you said) if the argument of linearity of sesquilinear form changes

posted on main :) .. math.stackexchange.com/questions/752038/… .. anyone any ideas ?

@EnjoysMath Let h be a sesquilinear form linear in the first argument. I think it comes down to why is h(0,x) = 0 and h(x,0) \neq 0 (not necessarily)?

@EnjoysMath Please, it is very important to me

@EnjoysMath Maybe you meant multiplicative identity...

@robjohn What an irony is to be deceived by own creation! :-)

@Chris'ssis It is a tricky question. The associated improper integral converges, but not the sum.

I was about to believe it converges ... hmmm

@robjohn Exactly.

@ParthKohli You are answering a lot of lhfs these days.

yas .. my rep is exactly 2014 !

@r9m nice!

@r9m Is it?

@r9m Due to me!

@Sawarnik Thank you ^_^

Do you have the intention that your rep becomes 2015 in the year 2015?:-)@r9m

:P

@r9m for your album

@SamiBenRomdhane that will be great ;)

@robjohn COOL :D

I wanted to watch the film "How I Came to Hate Maths" but I can't find an illegal download copy to pirate off the internet. Does anyone have any other suggestions for movies?

@Sawarnik didn't see that .. what was that ?

@Sawarnik how ??

@Ted I just stumbled across your math lectures on YouTube :D hahaha awesome!

@Sawarnik That's the only thing within my answering capability.

@r9m Isn't the inequality on your problem easy? Each term is smaller than 1/2.

@ParthKohli No you could do problems that are hard but not too advanced :D

@Sawarnik hell it is ... HELLFIRE :P LOL

@r9m Should I retract my upvote? :P

You don't upvote or downvote based on difficulty.

Or I think you shouldn't.

@Sawarnik as you wish .. how did you show each term is $\ge \frac12$ ?

So far, I've exploited this rule. :D

Such an idiot I am!

@r9m Sorry.

@Sawarnik if you want to remove the normalizing condition .. you can try to prove instead $\dfrac{a^3}{a^2+b^2} + \dfrac{b^3}{b^2+c^2} + \dfrac{c^3}{c^2+a^2} \ge \dfrac{\sqrt{3(a^2+b^2+c^2)}}{2}$

@r9m Means you know the solution? Just curious, no offense :)

Ok, let us see how to do that.

@robjohn Thank you for the opinion. I start to believe too that there is no known closed form. These kind of sums appear in the von Mangoldt function matrix.

@Sawarnik there are so many awesome people in Math.SE .. someone is bound to give an elegant solution to it :) ..

That comment is right... it really looks like something involving Lagrange multipliers.

@ParthKohli You know the stuff called Lagrange multipliers?

@Sawarnik A little.

Wow, you are so much more awesome that I thought!

@N3buchadnezzar I have tried it in Mathematica with NSum[] and it does not change when increasing Working Precision.

This is a very good introduction.

I'm trying to think of an elementary technique.

@MatsGranvik "Brothers"? Foes. :)

@ParthKohli Yup!

@MatsGranvik It's only an opinion. There may yet be, but I don't see it.

I haven't practiced that many problems with LM...

But you should see that introduction I sent to you.

I posted a question very important to me. please, any answer will be very very appreciated, thank you! math.stackexchange.com/questions/752080/left-and-right-kernel

@WillHunting Hi.

@Andrew: I hope you didn't find it a complete waste of time :)

@ParthKohli Read it! It was nice :)

@Sawarnik Olympiads don't accept L-multipliers .. thanks for adding the pre-calc tag :)

@r9m Olympiad question, there must be some nice solutions then.

maybe .. but both the source or official-solutions are unknown to me .. :(

But I don't like when olypmiads don't accept advanced things, after all they are nice tools, and if a student knows them well, why can't he use it!

@mirgee I don't understand your question. are you asking if it possible to choose one definition for both right/left sesquilinear forms ?

@GabrielR. No no! I know the definition has to change. I just don't know why!

@GabrielR. Let h be linear in the second argument. Is it true that h(0,y)=0 for all y?

I need to consider it at least as my PODW (problem of the week).

@TedShifrin Hi. A terminology question: Do you know when (and why) the horrible un-word "nullity" was coined?

@mirgee h(0,y)=h(0*0,y)=\bar{0}h(0,y)=0h(0,y)=0

@mirgee The thing is the concept of kernel is linked to (simple) linearity

@migree continued: since the kernel itself has to be a vector space

@GabrielR. Exactly. But how does not changing the definition of nullspace while changing the argument of linearity violate it?

@mirgee I see, sorry for doubting. There's actually no contradiction, you're right

No, @Daniel. But nullspace makes sense and it's only a small step from that to nullity. I use both words, sorry :P

@mirgee: I don't think it makes any difference. The conjugate linearity in one variable doesn't stop the kernel from still being a subspace. But one does need to specify left- and right kernel, as, without hermitian symmetry, they won't be the same.

Salut @Gabriel

@TedShifrin Null space is classical, no problem with that. But calling $\dim \ker T$ the "nullity" makes me shudder. It's worse than calling a primitive an antiderivative.

Salut @TedShifrin Do you speak French ?

I do that, too, @Daniel. You Europeans just need to chill!

Oui, Gabriel, bien sûr.

Un peu, je crois, @GabrielR.

@TedShifrin You 'murkins just need to stop inventing bad new words ;)

Hey, I have forgot some notation. While $E : F$ is the degree of an extension of a field E over field F, what does the notestion $G : H$ represent as far as group goes? The number of cosets?

J'ai même donné des conférences en français à Paris il y a longtemps ... et j'ai écrit il y a longtemps un bref chapître là-dessus.

Yes, @Andrew. And there's a formal correspondence thanks to Galois theory.

@AndrewThompson Yes. It's generally called the index of $H$ in $G$.

BTW, I use [E:F] and [G:H] :P

@DanielFischer vous aussi ? (ai-je le droit de vous tutoyer ?)

@GabrielR. @TedShifrin @DanielFischer Thank you for your efforts ;)

ça m'est égal, Gabriel ... :P

@mirgee: I assume that cool series question from yesterday is now settled?

Hi @TedShifrin

Ah, thanks Ted and Daniel, the word "index" came back to me know. Too many series-tests to remember in Calc, so I have a tendency to forget stuff..

@TedShifrin waouh ! des conférences dans quel domaine ?

I hope you are doing well this afternoon @TedShifrin

@GabrielR. Un peu, j'oublie beaucoup toujuors, mais si je vais en France en vacances, ca retournera. (Forgive the absence of the cedille.)

La géometrie intégrale, Gabriel ...

Heya @user127001

@TedShifrin You really liked it didn't you :D Well, not quite. I hope find a way to solve it without stirling (as you suggested) :P

Yes, my solution was totally without Stirling, @mirgee ...

Just the idea of the integral test

@DanielFischer héhé. Tu es Anglais ?

@Andrew: Whatever became of your differential geometry? Ahem. :D

@TedShifrin Ah... Will have to find in the older messages! Or would you remind me of your solution?

@TedShifrin je sais pas ce que c'est lol... est ce aussi exotique que la géométrie tropicale ? ;)

@Ted, have I asked questions in that direction? That could very well be, although I am not aware of it. We freshmen like to talk about stuff we don't really understand yet :)

@GabrielR. Non, je viens du département du Bouches de Weser, presque Français, mais cent-cinquante ans trop tard.

Non, pas si exotique ... Des formules cinématiques ... versions locales de Gauss-Bonnet et généralisations ...

oh, sorry, @Andrew ... I might be confusing you with another Andrew who was a senior.

Hi@TedShifrin I come now seeing your course on youtube and what made ​​me attention is the totally different method of presenting a course. I think you gave a prior written course and the session will explain the content of this course; I liked this method!

Haha, most likely :)

@mirgee: You want estimates for $\log n! = \sum_{i=1}^n \log k$ ... Now think integral test.

What do you have a course on at YouTube, @Ted?

@DanielFischer ok

They're videoing my (hard) integrated linear algebra/multivariable calculus/analysis course, @Andrew. I hope to get the first semester done next fall before I retire. But this covers multivariable integrals, inverse function theorem, intro to manifolds, differential forms and Stokes's Theorem and applications, and now we're doing eigenvalues/eigenvectors to finish up.

@Sami: Why do you say totally different method?

@TedShifrin j'ai oublié de te complimenter sur ton écriture: la plus agréable que j'ai vue pour un prof de maths !

(Encore un francophone ici ... Gabriel, tu connais Sami? :P)

Hm, looks like a lot of it falls outside of my courses. I do have eigenvalues/eigenvectors in both Lin. Alg. and Calc 2, though, but given your other topics, I assume that you introduce some more advanced applications.

LOL ... J'espère que ce n'est pas la seule chose d'intéressant, Gabriel :P

@SamiBenRomdhane @TedShifrin je sais qu'il enseigne en classes préparatoires !

Yeah, @Andrew. The first semester has all the rest of the linear algebra course in it (except for determinants this term).

Yes our method of presenting a course is to write all things from the definitions to the theorems and even the remarks on the blackboard:-)@TedShifrin

Ah, well I write plenty :P But I also try to get the class to suggest ideas, too, although I think I have been rushed this semester (thanks to missing 4 classes because of snow) and have been a little less relaxed.

@TedShifrin non loin de là lol. J'ai beaucoup aimé ton exemple sur le théorème d'inversion locale

Cool. By the way, I am considering taking Commutative Algebra, Real Analysis and Calc 3 next semester. Does that sound reasonable, or is it an insane workload in all three subjects?

Whoa, @Andrew. Commutative algebra is ordinarily a graduate course. What's going on?

Real analysis and Calc III is plenty.

Where are you a student?

Well, we are required to take three subjects a year, or we won't get a loan.

Bergen, Norway.

I could switch commutative algebra with discrete math, but I don't feel I have anything to learn from discrete math.

Oh, cool ... So I'm confused. You're taking beginning calculus, but you've already had a year of group, ring, field theory?

Ah, now I see the confusion. You're a different Andrew. @AndrewG was here earlier and he's the one I thought I was talking to ... AGH. :P

Hahaha

glares @user127001

:x

Maybe time to retire sooner than you thought?

Haha just kidding

Retire from here for sure, @user127001. Next year for "real" students. :D

REAL students

Go out with a bang

I'm teaching something I don't know in the fall, @user127001. Is that good enough?

Yes

Yeah, things got a bit messed up: our first semester, Calc 1 was the only mandatory math-course. In addition, a philosophy course and a programming course is mandatory for all math-students. (Philosophy is mandatory for all students.) For our second semester, I was to do Differential Eqn, Calc 2, Lin. Alg. and Abstract Algebra as an extra subject. I found out that I could drop DE without it affecting my degree, and since I hated it, I did. So now I'm considering commutative algebra,

which is an upper-undergraduate course here.

So how far does your abstract algebra course get, @AndrewT?

I am doing better in Abstract Alg than in Calc 2, though, but that's probably only because of my interests.

Sylow P-groups is the last section.

So just groups. No rings, fields, or Galois theory.

Yes, rings and fields,

no Galois.

Ah. I took commutative algebra as a senior (4th year) in college, but I found it very hard.

In particular, field extensions are heavily emphasized.

I should have dropped my algebra course the second we started doing Sylow's theorem the first day of class

I don't know why I waited until failing the first exam

So what exactly does the commutative algebra course cover?

@user127001, was there another section you could have switched to?

@TedShifrin no

@Ted, uib.no/course/MAT224

@AndrewT: That's a seriously sophisticated, advanced course. I would save it until you have a lot more math maturity. It's the beginnings of algebraic geometry. Do you not have a second algebra course to take first, that does modules and Galois theory?

You could do some number theory, some combinatorics. You should do a year of analysis and then some topology, etc.

My advisor here told me to never take number theory at our school

He said it's very dry and heavily discouraged it

Well, I am not a number theory fanatic, but most undergraduates love it.

And cryptology is number theory, too, with some computer programming thrown in.

Hi @JasperLoy

Yes, sadly, the Number Theory-community is dead at our uni, so it only offers a course once every few years. I could do Discrete Math, but I have kind of lost interest, it seems very simple.

I mean @Will

I can check out combinatorics.

@user127001 Hey Bart

@user127001: Well, my diff geo students may think I am as frustrating a teacher as you found Conway. I am not pleased with how my students are doing. Not pleased at all.

I'm sorry about your students' poor performance @TedShifrin

An intro discrete math course is too simple, @AndrewT. I agree, if you're handling your algebra course well.

You should be glad I'm not in your class or I would've brought down the class average by like 10 points probably

LOL, no, @user127001, I don't believe that.

@AndrewThompson All your course are taught in English ?

With 30 students, that's impossible!!

@user127001 Still haven't thought of a username?

See? I can't even do arithmetic

@GabrielR Norwegian if all Norwegian students, if one or more international students it is done in English,

unless it is a really big course with unmotivated students, then it is always in Norwegian.

Well, one of my students who should be an A student told me that the derivative of $2+\cos u$ is $2-\sin u$. His calculations became horrific. You'd think he would have stopped and double-checked?

@AndrewThompson it's cooler to do math in English lol

@TedShifrin Do you at least use a textbook for your course and assign homework problems? Or is that something you believe students should figure out for themselves

Haha, well, everybody blunders at times. Ted, here is the description for my Alg. course: uib.no/course/MAT220 Would you say its insufficient knowledge for Commutative Alg?

I had fun figuring out how to lecture on Grassmannians and Chern classes years ago in French :P

@AndrewT: That description includes Galois theory, including insolvability of the general quintic !!

Yes, its not updated.

It sounds like it needs to be a two-semester course. Where's the second semester??

They cut the subject several years ago.

Its one semester.

Growl. So there should be a second semester before commutative algebra. Someone over there is crazy nuts.

Or maybe they're calling it commutative algebra but it's doing more of what I'm saying. Who knows.

@TedShifrin I wonder how you could translate terminologies without the Internet lol.Bourbaki's books helped I guess :P

Hm, I can translate our Course-description for the Norwegian one.

Well, @Gabriel, I did ask my friends in Paris for occasional help on math words I didn't know.

How can you read Bourbaki

OK ... I'm outta here for now. A tout à l'heure, @Gabriel. Nice talking to you, @AndrewT.

See you, too, @user127001.

Thanks, same to you.

Later

@TedShifrin A plus tard

@user127001 is user127001 your original username ?

@GabrielR. no it's not my real name

What do you study? @GabrielR.

@SamiBenRomdhane Mp* info

@user127001 Is it the name the SE network assigned you ?

Ok is this what's known in france MPSI? @GabrielR.

@SamiBenRomdhane quite: it's actually the section that comes after you complete MPSI

hi

does anyone have a copy of Galois Theory by Edwards to hand?

I am evil ;)

@DanielF: This comes as no surprise to me.

I am deeply offended, @Ted.

I am evil ! I am Mojo Jojo !

Good afternoon folks

@GabrielR. Unlikely. 127.0.0.1 is the internal address of your local host

its a somewhat spacial set of numbers

:-D

@robjohn That a lotta down votes. Can you do it yourself?

@KevinDriscoll That's true... I may need some help ;-)

@robjohn The Crusade of Downvotes

@KevinDriscoll We should organize and Inquisition, too...

Nobody expects the Spanish Inquisition!

@robjohn Im not so sure I would survive such a thing...... I am a bit of an 'other' around these parts

hahaha, my roommate just showed me that video a few months ago

or rather he showed me a video of the sketch

@KevinDriscoll Knowing all the Monty Python sketches was a prerequisite for being a nerd when I was in college.