Given a convergent series of positive terms Σa prove that Σan^2 also converges. Is my proof correct? I am not sure whether I can manipulate the inequality an<ε into an<1. I can’t think why that’d be wrong but it just seems a bit arbitrary I guess.
#Series convergence proof
bleak sequoia
tender abyssBOT
Given a convergent series of positive terms Σa prove that Σan^2 also converges. ...
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hearty stirrup
Given a convergent series of positive terms Σa prove that Σan^2 also converges. ...
It looks like you've confused yourself by trying to use definitions.
bleak sequoia
It looks like you've confused yourself by trying to use definitions.
How so?
hearty stirrup
How so?
Well, you've invoked the test for divergence, which, as the name implies, is a test for... y'know, divergence. It's only conclusive when it concludes that a series diverges. And it looks like you perhaps might've intended to invoke the definition of a convergent series instead.
bleak sequoia
Well, you've invoked the test for divergence, which, as the name implies, is a t...
Wait when did I use the test for divergence?
hearty stirrup
Given a convergent series of positive terms Σa prove that Σan^2 also converges. ...
...literally the second and third lines.
bleak sequoia
Oh yeah nvm
Well yeah but it’s not to show divergence
I mean it’s called the test for divergence but
When a series converges an goes to 0
hearty stirrup
No.
I mean, okay, if the series converges, then a_n goes to 0 as n goes to infinity, but not the converse.
bleak sequoia
Yeah not the converse
But what I said is valid
So it follows that an<e
I’m only uncertain about the 4th and 5th line
hearty stirrup
I guess what you're saying is that there can only be finitely many n for which a_n < a_n^2.
I dunno, this whole proof is just out of left field for me.
bleak sequoia
I *guess* what you're saying is that there can only be finitely many `n` for whi...
Hmm yeah I guess
Or that for sufficiently large n an^2<an
bleak sequoia
You assumed `a_n < 1 ==> a_n^2 < a_n`.
Why is that false?
hearty stirrup
It's false when a_n < 0.
bleak sequoia
an is positive
hearty stirrup
Oh, okay.
bleak sequoia
Yeah sooo
Idk
Any other way to prove it maybe?
But I mean I don’t see what’s wrong here, doesn’t it follow that for sufficiently large n an<1?
hearty stirrup
I mean, the issue to me isn't quite so much that your logic is invalid as much as it is that a lot of it seems to go unstated.
Like, you didn't specifically mention that a_n > 0.
bleak sequoia
Like, you didn't specifically *mention* that `a_n > 0`.
Well the exercise doesn’t say that it says that Σan is a series of positive terms
But that’s the same thing right?
Oh wait
You mean that I didn’t state it in my proof?
So it’s incomplete?
hearty stirrup
I mean, it feels that way, kinda.
hearty stirrup
You mean that I didn’t state it in my proof?
That is what I'm talking about, yes.
bleak sequoia
That is what I'm talking about, yes.
Well that’s cool then cause I don’t really submit any assignments to anyone I only do them to learn so it’s ok if I skip something as long as I know it
But I see your point I would’ve definitely written this with more detail if it was an exam or something
hearty stirrup
Well that’s cool then cause I don’t really submit any assignments to anyone I on...
I mean, learning to be precise, explicit, and clear is very important when learning math.
bleak sequoia
Yeah I agree
hearty stirrup
If you skip over details in your written proofs, you could very easily also train yourself to skip over them in your mind, and so many promising proofs have been sunk by overlooked details.
bleak sequoia
This proof is definitely incomplete
bleak sequoia
If you skip over details in your written proofs, you could very easily also trai...
Yeahh ive also thought about that
But I simply don’t have enough time to explain every exercise rigorously
hearty stirrup
...why not? If you're self-learning, don't you have all the time in the world?
bleak sequoia
I am supposed to learn all this stuff in 6 months
bleak sequoia
...why not? If you're self-learning, don't you have all the time in the world?
I’m not self learning
Well I am right now
But
In studying cause I’m in uni lol
Physics
But our teachers don’t really give us assignments
They don’t care
hearty stirrup
...then demand your money back.
bleak sequoia
So yeah I’m basically self learning but with a deadline
bleak sequoia
...then demand your money back.
It’s free🤗
hearty stirrup
...wait, then how do they pay the teachers?
bleak sequoia
...wait, then how do they pay the teachers?
Well taxes
Most universities here are government funded
Well “people funded” I guess
hearty stirrup
Is there some kind of oversight committee you could report them to for their blatant negligence when it comes to teaching you?
bleak sequoia
Is there some kind of oversight committee you could report them to for their bla...
Nah that’s the way it works
It’s very different from school
hearty stirrup
No, no it isn't.
bleak sequoia
Idk how universities are in other countries
But here that’s how it works
You go to classes, they give notes, they post notes, you self study then exams
hearty stirrup
What is the point of teachers who don't teach?
bleak sequoia
That’s exactly my question too
hearty stirrup
If they're being paid to teach, and they're not teaching, then that's fraud.
bleak sequoia
At this point i have gotten more helpful information from discord and even chat got than my teachers
And I’m not kidding
bleak sequoia
If they're being paid to teach, and they're not teaching, then that's fraud.
Well yeah but that’s the system can’t do anything to change it
hearty stirrup
If you don't mind me asking, what country do you live in?
bleak sequoia
If you don't mind me asking, what country do you live in?
Oh I don’t at all lol, Greece
hearty stirrup
Well yeah but that’s the system can’t do anything to change it
The reason the system doesn't change is because everybody thinks that.
bleak sequoia
The reason the system doesn't change is because everybody thinks that.
True
hearty stirrup
Like, do you think the government intends to pay for a school where the teachers don't teach?
bleak sequoia
Like, do you think the government *intends* to pay for a school where the teache...
Not really
I mean that’s not the thing here
It’s just
University culture is just very different from school
At least here
hearty stirrup
It's not about "culture".
It's about people doing their jobs, that they are paid by the government to do.
bleak sequoia
It's about people doing their jobs, that they are paid *by the government* to do...
Yeah I totally agree with you
hearty stirrup
If the government finds out that they're not doing the jobs they are paid by the government to do, what will the government do?
bleak sequoia
If the government *finds out* that they're *not* doing the jobs *they are paid b...
They definitely know lol
It’s a well known fact
Their only obligation as far as I can tell is to attend the classes they have to
hearty stirrup
So then they do actually intend to pay for schools full of teachers who don't teach.
bleak sequoia
And teach for like x hours per week
bleak sequoia
So then they *do* actually intend to pay for schools full of teachers who don't ...
No teaching apart from lectures I guess
I mean some are definitely more eager to help than others
But generally it’s just lectures and notes that may or may not be posted online
hearty stirrup
Okay, I feel like I've seen you before, so I've probably directed you to Khan Academy?
bleak sequoia
Okay, I feel like I've seen you before, so I've probably directed you to Khan Ac...
You’ve seen me but you haven’t directed me there
hearty stirrup
It's basically an entire online math curriculum in the form of YouTube videos.
bleak sequoia
Yeah I’ve seen some videos from them I think actually
Honestly idk if it’d fit me
The way I like to study and the way I can learn is through books
And most importantly excercises
Plus there is always the problem that if I learn through some other source, like yt videos i might not cover the specifics that the class demands
hearty stirrup
I mean, can you not just read the syllabus and look up the specific topics?
bleak sequoia
I mean, can you not just read the syllabus and look up the specific topics?
Not really cause the syllabus is pretty broad tbh
Like for example
I happen to have a book for classical mechanics from another university
And they are very different
So it’s pretty specific stuff
hearty stirrup
I mean, they can't be different. Maybe in, like, notation, but if they're literally different in concept, then that just makes literally no sense. That would imply that there are literally two completely different sets of physical laws.
bleak sequoia
I mean, they *can't* be different. Maybe in, like, notation, but if they're lite...
They cover different chapters of mechanics
Like our book focuses a lot on relative motion while the other one barely talks abt it
Its just not good to work this way you have to work through the boom or through the teachers notes
Otherwise you don’t know whether what you learnt was actually taught by him, so you don’t know if you necessarily can use it in exams or you might have missed some stuff he thought was important so he focused more on it
hearty stirrup
Why does anyone even attend university in Greece? Like, literally just buy the textbooks and read them by yourself and you'll have a better time.
And probably learn better too since you can actually take the time to digest what you're reading.
bleak sequoia
Why does anyone even attend university in Greece? Like, literally just buy the t...
Well the degree of course
If you’re asking why they’re physically going to the classes
Then there’s different reasons, some teachers are actually good and teach you stuff
hearty stirrup
Okay, but then like, you can just do self-study until you understand the material, then enroll and ace all the classes and get your degree easy because you already learned everything.
bleak sequoia
Then there’s different reasons, some teachers are actually good and teach you st...
Some people just like going to college and drinking coffee together
Or do other stuff
bleak sequoia
Okay, but then like, you can just do self-study until you understand the materia...
Yeah but you’ll waste years
hearty stirrup
"Waste years" actually learning?
bleak sequoia
You can only give exams in up to 8 subjects per semester
bleak sequoia
"Waste years" actually learning?
Well waste extra years
hearty stirrup
I'm confused, which years are you suggesting are wasted?
bleak sequoia
The best thing to do is enroll and study on your own and pass the exams
bleak sequoia
I'm confused, which years are you suggesting are wasted?
Well you suggested you first self learn and then enroll
hearty stirrup
...except, like I just said, and like you said, if you do that then you're on a timer and you don't have the time to actually stop and understand what you're learning.
bleak sequoia
While I’m saying if you enroll and just self learn at the same time it’s more time efficient
bleak sequoia
...except, like I just said, and like *you* said, if you do that then you're on ...
I mean not necessarily, I could just ignore the exams, nothing is gonna happen to me I can take the exam for this subject next semester or the one after that or whatever
I know people who are in their final years and still haven’t passed calculus😭
In a physics university
hearty stirrup
Look, there's two distinct goals here. Learn the material, and get the degree. I'm saying doing one after the other is more effective in both goals than trying to do them both at the same time.
bleak sequoia
Look, there's two distinct goals here. Learn the material, and get the degree. I...
Look if I don’t understand the material I am not just gonna strive for a grade to pass the exam
Like even if I get a 50% I will reject that grade (I have the right to do that) and give the exam next semester
hearty stirrup
If you don't understand the material, it will be because you've put yourself in an environment which is not conducive to understanding the material.
bleak sequoia
If you don't understand the material, it will be because you've put yourself in ...
It’s mostly my fault I haven’t studied enough
But also the material is just too much
I have to learn ton of shit I’ve never seen before in about 6 months
Which is literally undoable
Like idk how someone could actually understand the material of the first semester in such a short time
I could whine about it all day long but it’s pointless
hearty stirrup
I mean, according to you they literally don't and it takes them until their senior year.
bleak sequoia
I mean, according to you they literally don't and it takes them until their seni...
For most people I know that’s the case
I’ve heard of some doing it though
Though I doubt they actually learn and I suspect they just pass the exams
Which I don’t wanna do
hearty stirrup
Or that they already learned the stuff.
hearty stirrup
Okay, look, for physics specifically, most of the calculus you're gonna need is derivatives and integrals.
hearty stirrup
Complex numbers are literally just numbers, though.
bleak sequoia
Yeah true
lol
I guess that’s not that important then
Well I also need to learn to work in polar coordinates
Which is hard af for me
Also very good linear algebra
hearty stirrup
Polar coordinates are simple if you remember Euler's formula, the proof for which is in differential calculus.
bleak sequoia
How are they simple they’re so fucking hard😭
hearty stirrup
Like, there's cases where you'd want to approximate a function with a series, but if you're a physicist it's not like you're ever gonna be short a computer to number crunch with.
bleak sequoia
Trying to describe motion in polar coordinates is hell for me
hearty stirrup
How are they simple they’re so fucking hard😭
Do you know Euler's formula?
hearty stirrup
e^(ix) = cos(x) + i sin(x)
hearty stirrup
That is polar coordinates.
bleak sequoia
It is?
hearty stirrup
...yes.
bleak sequoia
Damn I didn’t know that
hearty stirrup
x is the angle.
hearty stirrup
Right.
And then the magnitude is just the standard Pythagorean-derived distance formula.
hearty stirrup
I think you'll understand a lot better if you just understand the complex plane as exactly the Cartesian plane.
bleak sequoia
I hope so
hearty stirrup
Because it functionally is.
Both are exactly described by the Cartesian square of the real numbers.
It's just that points in the complex plane are numbers, so we can do arithmetic on them, while points in the Cartesian plane are points, so we can do geometry on them.
bleak sequoia
It's just that points in the complex plane are numbers, so we can do arithmetic ...
Hmm
Idk what this means
hearty stirrup
Hmm
Well, if I said to take the point (3, 7) and add to it the point (pi, -e), would that actually mean anything?
bleak sequoia
Nope
hearty stirrup
But if I said to add together 3 + 7i and pi - e i...?
bleak sequoia
But if I said to add together `3 + 7i` and `pi - e i`...?
Well I guess this kinda makes sense
hearty stirrup
I mean, it makes perfect sense. It's just adding complex numbers.
bleak sequoia
But we still could view imaginary numbers as points, couldn’t we?
hearty stirrup
I mean, yes.
I'm not disputing that.
I'm just saying that points in the complex plane are numbers, points in the Cartesian plane are not.
bleak sequoia
I'm just saying that points *in the complex plane* are numbers, points *in the C...
How are they not numbers?
I mean they are numbers
hearty stirrup
Well, if I said to take the point `(3, 7)` and add to it the point `(pi, -e)`, w...
This is exactly why not.
bleak sequoia
Like it’s a pair of two numbers
hearty stirrup
Like it’s a pair of two numbers
An ordered pair of numbers is not a number.
A number is a thing we can do arithmetic on.
bleak sequoia
An ordered pair of numbers is not *a* number.
Well ok but we could still describe points in the Cartesian plane same way we do in the complex plane
hearty stirrup
Well ok but we could still describe points in the Cartesian plane same way we do...
I think you mean we could describe points in the complex plane in the same way we describe points in the Cartesian plane, and yes.
bleak sequoia
I think you mean we could describe points in the complex plane in the same way w...
And vice versa
hearty stirrup
That is, complex numbers can be described as ordered pairs of real numbers.
hearty stirrup
And vice versa
No, not vice versa.
mental slate
hearty stirrup
Because the Cartesian plane exists to convert geometry problems into algebra problems and back again.
And then picked up the secondary application of graphing functions over real numbers.
To describe a Cartesian point as a complex number would be to talk about the number x + f(x) i, which, like, even if that made sense, why would you do it?
What interpretation of that makes it make sense?
bleak sequoia
No that doesn’t make sense ur right
hearty stirrup
A lot of the intuition of the Cartesian plane can help you with the complex plane - in particular it's also possible to describe Cartesian points in polar form - but the complex plane is like the Cartesian plane with additional properties that aren't really backwards compatible.
bleak sequoia
I see
Well anyways
Thanks for the help and the nice convo imma go to sleep. Thanks a lot byee
+close
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