@AsafKaragila I wouldn't normally, but the identity of the OP makes me think again.

@ZhenLin mathoverflow.net/questions/80581/very-general-topology

Would someone see a quick way to bound sum_0^infty exp(-C 4^-n) for some C > 0 in terms of C?

@HenningMakholm Yeah, I set up the stage. Now we wait for the jury and the executioner.

Hmm, maybe just Taylor it.

@Asaf: I remember reading Johnstone's paper, or at least the first few pages.

duplicate identified by arturo: math.stackexchange.com/questions/84741/…

Last call for 10k'ers!

And I'm spent. Goodnight folks.

Hello, can any help clarify a couple calc. II problems for me?

I can't figure out which one is wrong? Can any see anything that is blatantly incorrect? Thanks.

The first one is exp(x/6-1), and e^u's series expansion converges everywhere

So, you mean it should be (-inf, inf)?

Yeah, the 6^n in the denominator just makes it more convergent, not less. Maybe you got a denominator flipped in your ratio test?

or just left out the n!

ok, will input that and see if that works? ;)

There is still another incorrect one besides #1.

Now, #3 certainly doesn't converge in (0,12). n! grows faster than any power function, that's why exp()'s series converges too.

And lastly for #4 this is the geometric sum u^n for n=1 to inf where u=(x/6-1). Now |u|<1 implies what for x?

(#2 is alright)

Since they are all power series, the root of the linear expression being raised to the nth power must be in the center of the convergence interval. That alone eliminates a lot of possibilities.

How do I know they are in the center?

The only remaining doubt is matching {1,4} to {B,D}.

the center of the interval will be the x that makes all the terms 0

@Mr_CryptoPrime Basic theorem about convergence of power series. The disk of convergence is centered around the development point.

@HenningMakholm Sorry, I am not really acquainted with terms such as "disk of convergence". This is actually an online class, so the only way I can learn is basically through online math videos and resources such as this...lol

disk is when you look at the complex plane instead of just the real line

@tb: I noticed you left me some "homework". I don't like unfinished things.

@Mr_CryptoPrime Don't you have a textbook?

(although you can also get abscissas of convergence, but these are like generalized disks)

@Matt Yes, I just noticed :)

@HenningMakholm Yes, but I never understand half the things they say...I usually depend on patrickjmt.com for everything.

@tb: I noticed that you have been answering almost all of my questions, ever since I joined SE.

I don't really get the point of video lectures. They're impossible to skim quickly if you need to refer back to something, and waiting for someone to speak everything is much slower than just reading a transcript.

And I don't get the point of lectures.

Totally agree with Henning

I agree with Henning and Matt.

@Matt I remember that there were quite a few questions of yours that I answered (or commented on), but almost all seems to be exaggerated.

Or video lectures. Both is totally boring just like watching TV. Except for you can't pause when you need to go to the toilet.

@Matt Well, at least there is the theoretical possibility of interrupting the lecture with clarifying questions if you're actually in the same room.

Haha, I don't like lectures, but videos allow you to pause and rewind at your whim.

But I did skip most of the lectures when I studied, because they were just reproducing content from the textbook that I could read at my leisure instead.

@HenningMakholm: Haha. Our lectures are all completely silent. You know why? Because lectures are for lecturers to show off how well they can do stuff they have been doing all their life. By writing so fast that you get a cramp in your hand and can barely keep up with him.

I just wish college didn't go so fast, they expect me to learn like 10 different convergence tests in like 2 weeks...

You can pause and reopen pdfs at whim, and you can also skim them infinitely easier.

@Henning @Matt Did you teachers get mad at you?

Huh?

Why would they?

Think you are not listening

@Matt That is exactly how I feel...I don't go to physics or chemistry lectures anymore because they are like that. Geology is good though.

And therefore bad

There are 300 people

They don't notice and they don't care

Actually, I fell asleep once in the first row due to major lack of sleep the previous night and that time the lecturer apparently did notice...

or so I'm told.

What a waste of time.

Ah, I remember in hs I used to feel like I was letting teachers down if I wasn't doing well.

waste of time is an understatement.

@MrCryptoPrime: I've stopped going more or less, too.

Also in hs, one guy slept in the front row like every day. Whenever he started snoring it was my official duty to kick his desk to wake him.

Good for you. Three of my teacher actually punished me for skipping or not listening many of their lectures by giving me low grades.

Besides, I don't understand the idea of copying a book to the blackboard and then back onto paper while being disturbed by talking of others and sleepy because I have to get up early.

Two of them gave no explanation.

I got 95% on my last Calc. II exam and 32% on my last physics exam (the average was 35), it's ridiculous how capitalistic and apathetic college has become.

In a university organized for adults, the point is to give the students various possibilities to learn the subject matter, and then afterwards offer exams that allow the students to get a certificate that they did learn it. Sitting in on lectures is an offer, not something that benefits the lecturer. Why should he care that the students use other offers to learn from?

@tb: I didn't count them. But it looked like a lot.

@HenningMakholm Yes, but it is frustrating that they force you to pay for something you are not going to use, such as worthless lectures of rambling professors.

I didn't mean to be persnickety, sorry :<

@anon Just ignore his rudeness, then he becomes quite alright.

@Mr_CryptoPrime I didn't pay anything. There were people doing that for me. (I.e., taxpayers)

@MrCryptoPrime: I think "rambling professors" is too general. Some of them are actually quite nice. If you say it like that it makes me feel sad for the very few nice ones.

@HenningMakholm Oh, damn...I pay $4,000 USD every semester (all by myself). A job inexperienced 20yr old in the big ruthless world trying to survive haha

And I'm quite sure that if the university could get a larger percentage of students to pass by using their resources on something other than lectures, they would. They're paid per course passed per student, after all.

@HenningMakholm: Thanks for letting me know there are other rational people who think the same. Actually, in many universities (US), the size of a class and the attendance rate might say something about the quality of teaching of the lecturer. So they punished me for their own interest.

@Matt No, I said I went to all my geology lectures above. I really do like that professor and several I have had in the past, but there have been a lot of really crappy ones in my opinion who are really inconsiderate of us.

For those fresh-graduated lectures/professors, I can understand their urge to gain tenure. But for those tenures who have even been a dept chair before, I don't understand why they need to care about their faces so much.

@Tim: I don't think it has to do anything with tenure.

It varies a lot whether being a department chair is something to be proud of. At some places it just means "once lost the draw among the tenured faculty".

Then faces maybe

Ok, sorry got distracted. Back to that homework problem...for #3, I used the ratio test and got |(8x-6)/6|*lim|n+1|. Assuming I did that correct, this would imply that it converges to (-inf, inf)?

@Mr_CryptoPrime Can't be right because the (-infty,infty) obviously belongs to series 1 (where every term is less than the power series for e^{x-6}).

I don't think the answers are one to one?

Hm. Didn't even consider that they might not be.

They have been onto, in the past, so you can't always assume it haha

@Mr_Crypt: Two points. (1) You can't pull the constant out of the limit if the constant is 0. (2) Otherwise, you pull it out and you get infinity, so the ratio test is failed.

Oh, never mind, I just got it! :)

I am confused about what they mean in this question? Are they saying "a_n" or "a*n" or something different?

take the power series for exp(x). replace each x^n with x^(2n+1). Then think about it.

How do I know to replace with x^(2n+1)? (in dumb people terms please)

replace x in x^n with x^2 and you get (x^2)^n=x^(2n). now multiply by the outside x and you get x^(2n+1)

Good night folks!

Be seeing you!

oh, ok, yes I think I tried that, let me find that page I wrote it down. Bye Matt.

also, putting the responsibility of checking answers to computers sounds like a bad math class

Haha, ya...welcome to the 21st century?

so I have sum[n, inf](x^(2n+1)/n!)

yes. now how you have to input that I'll leave up to you...

Okay, I have to go too. Bye guys.

Well, I managed that far by myself prior to inquiring on here...this is where I am confused as what to do? :(

bye, Henning, Bye Matt.

OMG, nevermind, I just understood what I have to do...wow!!!! Talk about overthinking the problem, I feel like such an idiot! Oh well...lol

Ok, I just have one more, then I think I will be able to do all the rest. Upload it in just a sec.

Using the ratio test I got: |x/9|*lim|(n+2)/(n+1)| = |x/9|*1 = -1<x/9<1 = -9<x<9. That obviously does not help me? What do they mean: "find a formula". A formula for what? The nth term?

The "request for leave" application gives me a lot of 404's...

Holy crap! They wrecked the user info pages!

How am I supposed to find my answers efficiently now, when I have to switch between a dozen tabs?

By golly!

Good, so it's not just me. (I'm not here. Leaving again).

@tb I just noticed that.

Well, the same info is there, but what required some scrolling before now needs three or four clicks.

It's empty in here. Just 3.

Hmm. I just refreshed and I now see 6.