How to study Konrad Knopp book on infinite series?

[elliti123]

I am currently self studying the book and I think it is a great book and it really does go deep into the subject. But about how to study this book i am trying to prove every theorem that i come across and it is new. To get the most out of this book i would like to know other opinions on how to approach it. Thanks.

 

[lavinia]

I am currently self studying the book and I think it is a great book and it really does go deep into the subject. But about how to study this book i am trying to prove every theorem that i come across and it is new. To get the most out of this book i would like to know other opinions on how to approach it. Thanks.

I don't know the book but working out examples is always instructive. I would try to categorize the theorems into an architecture to get a big picture of what the theory says.

 

[jedishrfu]

With any self-study, you want to use your time effectively and not get bogged down in detail. Perhaps, it would be better to read the chapter again after having completed a few problems and not do every problem in a linear fashion.

Some books are setup with tracks where the student goes through all chapters doing track 1 reading and problem solving and then go back and do a second reading for track 2...

For your book, you could read and outline the chapter then use your outline to solve a few problems then go back, reread the chapter and refine your outline and do more problems. Also it wouldn't hurt to digress to other books when you're stuck on a problem or proof. In the case of proofs, there are often multiple ways to prove things and doing the digression you'd learn these alternatives.

 

[elliti123]

Yes indeed i do actually refer to other sources on any detail that i find that does not appeal or i do not understand it. Just one note about the book, it is basically full of theorems: theorem after theorem sometimes maybe sometimes there examples and explanation about the definition or any detail that needs to be pointed out. So overall it is kind of dry so i thought of teaching myself analysis side by side too. Although i have tried the spivak chapter on limits it really really helped me to actually grasp what it actually mean for a sequence to be null. But do i need to be able to prove every theorem on my own when faced at first sight ?

 

[verty]

Yes indeed i do actually refer to other sources on any detail that i find that does not appeal or i do not understand it. Just one note about the book, it is basically full of theorems: theorem after theorem sometimes maybe sometimes there examples and explanation about the definition or any detail that needs to be pointed out. So overall it is kind of dry so i thought of teaching myself analysis side by side too. Although i have tried the spivak chapter on limits it really really helped me to actually grasp what it actually mean for a sequence to be null. But do i need to be able to prove every theorem on my own when faced at first sight ?

No, you don't need to prove every theorem. If the proof is given, you just need to understand it. Once you understand it, you will believe it. Think of the proof as a reason why the thing is true. If you have the proof in front of you, you just need to understand it.

 

[lavinia]

Yes indeed i do actually refer to other sources on any detail that i find that does not appeal or i do not understand it. Just one note about the book, it is basically full of theorems: theorem after theorem sometimes maybe sometimes there examples and explanation about the definition or any detail that needs to be pointed out. So overall it is kind of dry so i thought of teaching myself analysis side by side too. Although i have tried the spivak chapter on limits it really really helped me to actually grasp what it actually mean for a sequence to be null. But do i need to be able to prove every theorem on my own when faced at first sight ?

It doesn't hurt to try but only to understand why the theorems are true.

 

[verty]

I always found it infeasible (in the sense that it's quite difficult) to prove theorems because usually there's a trick involved and finding the proof is like finding the trick. I'm good at finding tricks so I can usually find a proof but it's there in front of you and written up nicely, so I prefer to just read it to see what the trick is.

 

[lavinia]

I am currently self studying the book and I think it is a great book and it really does go deep into the subject. But about how to study this book i am trying to prove every theorem that i come across and it is new. To get the most out of this book i would like to know other opinions on how to approach it. Thanks.

I glanced at the book on line and it seems difficult to read. Perhaps you could use companion books to make things easier.
Alfors book on Complex Analysis has a chapter on infinite seres in the Complex plane. Also Courant and Hilbert's book on mathematical methods in Physics covers series.
Fourier series can be studied in an analysis book. I will try to think of others.

Also try doing the exercises at the end of the chapter first then research backwards to fill in the holes in your ideas.

 

[verty]

There is a book by Bromwich from 1907, "An introduction to the theory of infinite series", which is so well written that I believe it has never gone out of print. You could always look at that (find it for free on archive.org as well).

 

[elliti123]

Thanks for all the help, i looked at the book from Bromwich it seems rigorous and beautifully written i will try to study that as well.

 

[elliti123]

But do i need experience with calculus for this book to continue on with it ? Because i feel like i do understand everything and how to use them and even the proofs

 

[verty]

But do i need experience with calculus for this book to continue on with it ? Because i feel like i do understand everything and how to use them and even the proofs

Okay, I think I misunderstood what your question was. I thought you were saying you were struggling with the book because you were trying to prove every theorem and it was difficult. So I replied that I think proving theorems is infeasible in general because if one proves every theorem without looking at the proofs that are given, it'll be very difficult and will take a lot of time. If the proofs are given, surely the author means you to read them; that is what I was saying.

But I see now you weren't actually saying you were struggling. It sounds like you can just continue because you understand everything, so I guess it isn't using things that one is assumed to know. So it sounds like a good book and it sounds like you are doing fine with it so far, I would just continue.

I liked Lavinia's first suggestion, that studying the examples is very important and seeing the theorems as some kind of large architecture is helpful to understand how they all fit together. And to do that, reading all of it in one book is nice, if you understand it all and aren't having problems.

If you did try to pick it up from separate books, it might be more difficult to see it all together because each author would cover it a bit differently. So I do recommend continuing with this book, seeing as you seem to be enjoying it.

But if you do find that you need to know analysis first then by all means defer learning this material till you learn the analysis stuff.

And if you did need to know things from analysis, companion books would be a way to just get what you need.

 

[elliti123]

Yes sorry if i pointed out my idea wrong and yes i seem to get everything since i am planning to try to go from mostly analysis into calculus other way around or in other words to build my mathematical maturity before i go into calculus thought in high school since i do not have any strong background in calculus even though i studied it two years ago but i left it because of infinite series part that really frustrated me and i could not get my head around that.

 

[verty]

Yeah, just keep going and also look at Bromwich because it was written at the time when that math was very popular so it may have good insights as well.

 

[elliti123]

But why would this kind of math be more popular at that time?

 

[jedishrfu]

Throughout history, some math is more popular than others. During the Greek era it was Geometry then centuries later Algebra came into ascendency and then later still Analytic Geometry. It tracks the expansion of knowledge and the great trends in thinking.

An analogous one in physics is the popular interest in Newton then Maxwell then Relativity and QM and more recently String theory. At least it seems that way.

Math timeline

http://www.snipview.com/q/Mathematics_timelines

 

[lavinia]

Throughout history, some math is more popular than others. During the Greek era it was Geometry then centuries later Algebra came into ascendency and then later still Analytic Geometry. It tracks the expansion of knowledge and the great trends in thinking.

An analogous one in physics is the popular interest in Newton then Maxwell then Relativity and QM and more recently String theory. At least it seems that way.

Math timeline

http://www.snipview.com/q/Mathematics_timelines

I think this is true for sure in Mathematics. Chaos theory went mad popular in the 1980's and 1990's. Now it's glow has faded.