howdy @robjohn

@TedShifrin nice to see you again.

@TedShifrin Oh, that's a hobby.

@TedShifrin I think I will probably stay single. Once you are past 30, it's hard to find someone

@Mike whining is?

well, Jasper, I did ... but then I lost him ...

I just answered a lhf and then the question changed!

@WillHunting it is harder, but not impossible.

Now I have to delete my answer

@TedShifrin Of course.

@WillHunting I hate it when that happens.

It's pretty much an avocation for you, @Mike.

@robjohn I will have to join a dating site. I tried okcupid before

Well, I answered a question and the guy's gone mute for 24 hours.

I refuse to write out a complete solution ... but my answer and the ensuing comments pretty much do it.

@TedShifrin Oh I did not know about that. You broke up recently?

LOL, no, Jasper ... a long time ago ...

Another downvote on what I think is a good hint! >8(

I wish these people would say what they think is wrong.

Well, "good" is in the eye of the beholder. But an OP downvoted me because I refused to write a whole textbook as an answer for him.

@WillHunting I just answered that question.

My answer ended with "who cares"

I know I don't care.

@user127001 I recommend you join okcupid.com, best dating site I know of, you get to send messages for free!

Apparently the OP liked "who cares" so much they accepted my answer

@TedShifrin This was so simple, I figured it was a beginning course homework question, so I only gave a hint.

I approve, @robjohn. I almost always give just hints as responses to such questions.

And then someone comes along and gives a complete solution. Makes me livid.

So I'm hardly answering anything any more.

@TedShifrin I gave an extremely detailed answer yesterday and got very few votes, so you never know what to do.

well, @robjohn, at this point in our lives, we hardly need votes.

Why would one care about this?

@Mike: I don't even care to look.

Is there an easier book than Gallian for learning intro abstract algebra?

What do you want from a book?

Gallian and Fraleigh are bottom of the line, @user127001. My algebra book requires more thinking, less symbol pushing.

@TedShifrin certainly. However, as a moderator, I see a lot of this going on, and so it bothers me in general. In general, I don't like to see good answers downvoted because it gives the wrong impression to people looking for answers later. I don't really care about the 2 points lost.

yeah, @robjohn, I get your point. I'm just frustrated by a lot of things here ... and I just hit my one-year anniversary two or three days ago.

@user127001 You should also ask yourself: do you want rings to be defined with or without 1? when choosing a book

I prefer very thick textbooks that go into unneccessary detail for anything that's even trivial during their explanations of a concept and in the worked out examples

Jasper, for a first course, with $1$ is JUST fine.

ugh @user127001

the antithesis of a good book

hides from @Kevin

@TedShifrin He's gonna ask you how to solve some horrifying integral.

@TedShifrin Right, but nowadays about half the books in the market have 1 and half without 1.

@WillHunting Who cares?

LOL, @Mike, I doubt it.

Jasper, I'm one of the rare ones who did rings first, too, and I stand by that for a book for non-professionals.

@Mike It is too irritating for me without 1

@Mike @Ted You know I would just to torture chat. But htankfully I figured out how to solve my horrifying integrals.

@TedShifrin My algebra course did rings, then groups, then fields, then modules.

We forget that 90%+ of math majors aren't ever going to grad schools. But math departments still have their heads in the sand and pretend everyone is.

Cool, @Mike. I do rings and field extensions first semester, groups and Galois theory (and projective geometry) second. No modules.

Better math books are dense and minimal?

You have to learn to think and work things out for yourself, @user127001. Somewhere in the middle is my ideal.

What precalculus and calculus books do is NOT what a good book should do at the higher levels.

And even calculus students can't do a problem if there isn't an example to copy. Sigh.

I like books which are extremely concise

When I say fields, I mean Galois theory.

We had a maybe two-week stint into modules so we could decompose them.

Right, @Mike. I introduce field extensions quite early.

Yes, I love modules, but sacrificed them for my text ...

Unfortunately I can't afford to buy books anyway so I illegally pirate all of mine from internet downloads

Well, it appears your mother destroys all your books, @user127001, so that's wise.

@TedShifrin I frequently get complaints from students that the problems we ask the students to solve on assessments arent always the same as the problems from homework/lecture....... my response is just kind of "DUH!"

I still prefer by a large margin to hold a hardback book in my paws.

@user127001 I like Cohn's Classic Algebra

Yes, @Kevin. The sad thing is that even when I tell my best Honors students in my multivariable math class that I'm doing an example JUST like a homework and JUST like a future test problem, they STILL ignore me.

I prefer a physical copy too, but a tablet is as close as I'll get

Heh...... I get that too

For years (and in my book) I have done an example of a horrendously complicated parametrized path with a nonobvious vector field and told them to compute the work. Of course, the field is conservative, and I've clued them in to this. But every year someone tries to use the parametrization to compute the work on the exam and writes a page of garbage ... with no answer and 20 wasted minutes.

@Ted I legitimately can't read Matt books on a screen. It's painful. It's even worse when I need to flip around, as I often do.

I don't argue.

I don't even like reviewing manuscripts that way, but that's the way we get 'em these days, and I'm not going to waste hundreds of pages of paper printing them out.

When I graduate and move out I will start to buy regular books again

well, @user127001, I'm trying to get rid of most of the books in my office, so hurry up.

If I go to grad school at UGA we can save on shipping costs

I'm not going to get all of them? :-(

LOL, @user127001 ... I'm not sure I'll be around long enough.

You coming to collect 'em, @Mike?

Ya I think theres still some legitimate advantage sto physical books. its so cumbersome to 'flip through' a digital book.

It's very sad to think of getting rid of some classics that I've had for 30+ years ...

yeah, @Kevin, it sucks.

Then don't!

OK, time to go cook dinner. Have fun ...

@Ted Ok, I'll be right over.

Okey dokey @Mike.

@Ted Did you see how I solved my integrals?

@KevinDriscoll You should search for the user Cleo and see how she computes integrals lol

@WillHunting Oh I see. She just drops the answer and doesn't explain how it was found.

Hence the quot by Gauss about scaffolding

@KevinDriscoll I think she is Ramanujan reborn

Ah crap

I think I will go to bed

Night @JasperLoy

@user127001 Remember to join okcupid.com, lol

@WillHunting Her answers are completely useless.

@PedroTamaroff I wonder if she is a genius or a troll

Well, not of all them.

But most of them are just the result.

Who is this "she" you guys speak of?

@user127001 Do you have a female relative?

@user127001 Cleo

@PedroTamaroff I probably do have some female blood relatives, but I don't know them

@robjohn. Thanks for looking at my question. So it is the partial sum?

@eXtremiity one place or another you need to have a partial sum.

That does make more sense. People have been using $f_{k}(x)$. And I've always seen a problem with that.

@eXtremiity either $f_k$ needs to be a partial sum or the limit needs to have a partial sum

Ok, I'll go with the former.

Now it gets very very tricky.

Can you go through this with me

@eXtremiity well if $f_k(x)=\sum\limits_{j=1}^k\dots$ then you can use $f_k(x)$ in the limit

$\underset{n\to\infty}{lim}\left(\int_{-1}^{1}\left| \sum_{k=n+1}^{\infty} f_{k}(x)\right|^{p}dx\right)^{\frac{1}{p}}$ @robjohn. Is this right?

@eXtremiity yes, but you don't need the exponent of $\frac1p$

Why not?

@eXtremiity and that equals $0$

I thought it was part of the equation I had to prove.

Hey @robjohn I'm about to resort to posting a full question, I'm stumped on some integrals

Oh.......take both sides to the power of p ?

if $\lim\limits_{n\to\infty}x_n=0$ then $\lim\limits_{n\to\infty}x_n^{1/p}=0$

Ok I see. Next problem I have, is where you can put the limit.

You have put the limit...inside the integral or?

Because I know that you can't put it inside, only under certain conditions. Or does this not matter @robjohn.

@eXtremiity Of course, you might want to use Minkowski's inequality and then you might want to keep the exponent

Ok that is pretty !

@MickLH at first sight, I would be surprised if there was a closed form for that.

@eXtremiity $L^p$ is a metric space and follows the triangle inequality. (Minkowski is essentially the triangle inequality for $L^p$)

Where exactly do you situation the limit in$ \underset{k\to\infty}{lim}\left(\int_{-1}^{1}\left|\sum_{k=n+1}^{\infty}f_{k}(x)‌​\right|^{p}dx\right)$

Because to come to an answer of 0....does the placement of where the lim is important?

I am under the assumption that you have put the limit inside of the integral.

@eXtremiity since you are trying to show that it is $0$, you can simply use $$\lim_{n\to\infty}\sum_{k=n+1}^\infty\left(\int_{-1}^1 |f_k(x)|^p\,\mathrm{d}x\right)^{1/p}=0$$

Oh no. I'm seeing things that I have not learn't.

Oh wait...but that's greater than the original equation, is it not?

We're doing a comparison of a sort?

Oh, I don't know :(.

@eXtremiity Minkowski says that $$\left\|\sum_kf_k\right\|\le\sum_k\|f_k\|$$

Ok.

Ok so I see how you got your previous equation. Yet I still don't see how it is equal to 0.