Final Exam Review, Due December 10

• Which topics and theorems do you think are the most important out of those we have studied?

I think the most important topic is, frankly, how to prove things! The different methods that can be used (direct, contrapositive, contradiction, induction, etc.) as well as the basic structure of how proofs should look (logic, quantifiers, assuming the premise, etc.)

Also, the whole concept of sets was a relatively new topic for me, and pretty much made everything in this course possible.

• What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

- How to prove that things are unique. (For example, the intermediate value theorem, proving that there exists one root in a certian interval)

- Review well-ordered vs well-defined and when we use these. 

12.4, Due December 8

Difficult: Well, I do understand how to use these theorems. And I suppose I understood what was going on when they demonstrated the proofs, but on the division and multiplication one, the lemmas they chose to construct were quite.. overly convenient. Like I have no idea how they would do it, they would just prove random stuff, and then it all magically turns to epsilon!

I guess it would be helpful to help understand why and how they would bound f(x) and g(x). Maybe we could discuss what it means in general to have things "bounded".

Interesting: Like I said, it is really interesting and cool to use these proofs once you get through the gross epsilon-delta proofs. But I guess that's what we're here to learn, the gross stuff that makes it all work!

12.3, Due December 5

Professor Jenkins, I had this blog post typed last night, but I guess I accidentally saved it as a draft instead of publishing it. I apologize, but don't worry about changing it in the gradebook, since it was my mistake. 

Difficult: Holy cow, this section was a whole lot to take in!! Given I've never dealt with epsilon before, it was quite easy to get lost in the Greek letters. I understand the logic of what we must prove (I was able to write it using quantifiers! Yeah!) However, the biggest difficulty for me was figuring out how to choose each delta. It makes sense for linear expressions, but once we get to quadratic, polynomial, and rational expressions, I got lost where we were going.

What do they mean by upper bound? They never clarified what that was exactly referring to.

Interesting: This is really interesting, because in my Calculus AB class, to find limits we just looked at graphs and said the limit existed. But now we can prove it!! I feel pretty cool when I win at these.

12.2, Due December 3

Difficult: Awesome, so I understood why we use sequences of partial sums, but will we always need to prove two lemmas to prove convergence? The first lemma to prove {s_n} can be expressed in terms of n and the second to prove that the limit is convergent to a limit? I guess once we have those, we just plug those limits in and the proof is pretty easy.

And, holy cow, it looks tricky to prove a sequence diverges! Maybe it was just because the harmonic sequence is specifically difficult to prove, But those algebraic manipulations didn't seem intuitive at all. Is there a general structure for a proof that a series is divergent? Or does it always just depend on what the series is?

Interesting: Well, I've always used the words series and sequences interchangeably, but now I know the difference! A series is just the numbers in a row, where a series adds them all together! I've also hear of the harmonic sequence before, so it's really cool to finally know what it is. I can't believe it diverges, it seems like it wouldn't! Oh well, my intuition has definitely deceived me before.

12.1, Due December 1

Interesting: Woah, this is a way cool concept! I never thought of a series as a function from the natural numbers to the real numbers. Also, it makes sense that they would have to make a definition for limit, since just getting infinitely close isn't specific enough. Cool calculus proofs and concepts I have never seen.

Difficult: I understand most of the concepts, though I feel epsilon is really important so it'd be good to talk about that one again. I also would have a really difficult time constructing a proof from scratch: why do we need the ceiling function? I also noted that sometimes we assume "for all" and other times we assumed "there exists", so it'd be helpful to discuss the logic behind proving a function is convergent or divergent, or what it converges to.

Question Responses, Due November 25

• What have you learned in this course?

Goodness, what haven't I learned?? I have learned a lot of mathematical notation, logic, and a lot about how mathematicians are able to say the things they do. Things that I once considered trivial, like whether an integer was even, I can now prove in a heartbeat! I have learned how to prove things in a logical and complete sense, and how to prove things that I didn't actually believe were true. (I still can't believe there's only one prime number that's one less than a cube!) I learned a lot about what functions really are, which is pretty cool considering I've been using them now for four years; it's about time I learned what one is! This has also been my first exposure to number theory, which is way interesting and exciting.

• How might these things be useful to you in the future?

Being a math major, I honestly don't know how you would progress without this class. I really did feel like it was an excellent transition into a higher level of mathematics, and now as I take Math 113 and up, I will definitely be able to look at it in a more comprehensive and enlightened way.

11.5-11.6, Due November 24

Difficult: Could you elaborate more on proving the fundamental theorem of arithmetic, specifically the part proving there are unique factorization?

Wow, there were just a lot of theorems and corollaries proved (Eleven new ones in fact!) I followed each proof pretty well, it's now just a matter of making them intuitive and remembering them. And then, of course, learning how to use them. =)

Interesting: I really think it's interesting that the "Fundamental Theorem of Arithmetic" is that all numbers can be expressed as products of primes. What makes it so fundamental? I mean, we've gone five sections previous to it without actually having to prove it. Anyway, just an interesting question I thought of.