# Recent content by Dr. Seafood

1. ### Inductive proof in complex arithmetic

Actually the formula is \cos x = \frac{e^{ix} + e^{-ix}}{2} but the general idea of using this formula was very helpful! It's a lot easier to show this directly by plugging in that formula and using the geometric sum formula, than to use induction. Cool problem though! Thanks guys.
2. ### Inductive proof in complex arithmetic

Oh yeah! I saw it before in first year but it's been a while since then, lol. That's not really too in-depth but the only stuff I've done with complex numbers was studying inner product spaces, and all we didn't need stuff complex exponential notation to do that. But yeah I'll try that formula...
3. ### Inductive proof in complex arithmetic

Which complex notation are you talking about? We actually haven't learned much about complex numbers in particular ... It's an analysis class, so the prof defined \mathbb{C} as \mathbb{R}^2 with "funny multiplication" and then we immediately started talking about Cauchy sequences, topology...
4. ### Inductive proof in complex arithmetic

Sorry sorry sorry I mistyped the question! Certainly summation begins with k = 0.
5. ### Inductive proof in complex arithmetic

Homework Statement Prove that for any n \in \mathbb{N} and x \in \mathbb{R}, we have \sum_{k = 0}^{n} {\cos{(kx)}} = \frac{1}{2}+ \frac{\cos{(nx)} - \cos{[(n+1)x]}}{2 - 2\cos {x}} Homework Equations None I can think of. The Attempt at a Solution Try induction. The result holds if n = 0...
6. ### Non-convergent power series but good approximation?

^ Power series are just polynomials, right? So essentially you're approximating using low-order polynomials. I don't know anything about QM but that'd be my guess as to why what you're doing is reasonable.

8. ### Integrals on arbitrary (bounded) domains

Homework Statement Let A = \{(x, y, z) \in \mathbb{R}^n : 0 \lt x \leq 1, 0 \lt y \leq 1 - x^2, 0 \lt z \leq x^2 + y\}. Define f : A \rightarrow \mathbb{R} by f(x, y, z) = y for each (x, y, z) \in A. Accept that Fubini's theorem is applicable here. Find \int_A f. Homework Equations Fubini's...
9. ### Apostol definition of interior point and open set

Hopefully this will clear some stuff up. Let S \subseteq \mathbb{R}^n. A point x \in S[\itex] is called an interior point of S if there exists a number r > 0 such that, whenever a \in \mathbb{R}^n is such that \Vert a - x \Vert < r, we have a \in S. To internalize this definition, let D_r(x) =...
10. ### Proving the proof by contradiction method

^ Yeah, I know about that stuff!! That's a super good idea, thanks a lot. I will upgrade my presentation to discuss transfinite induction. Thanks for the idea!
11. ### Proving the proof by contradiction method

I will show you what I have prepared on induction. My goal is to give new light on well-understood ideas, especially for freshman and juniors. Everyone learns mathematical induction in first year, so I just want to show a less common characterization of this concept. Let \mathbb{N} = \{1, 2, 3...
12. ### Proving the proof by contradiction method

I realize this. I'm presenting to undergrads, from freshman to seniors. Myself, I'm in my 2nd year. The only "axioms" I want to use are just simple intuitive things like that sets exist, that "set membership" is well-defined, that I can take unions, etc. Of course, considering my audience, I...
13. ### Does your school's math curriculum satisfy you?

The problem is that these new ideas are presented in the lecture. They're in the book too though. The theorem is drawn immediately after the definition is made, giving me no time to get intuition for a new concept and so I end up just scribbling down everything so I can read it later. It makes...
14. ### Does your school's math curriculum satisfy you?

Sometimes I feel like if I didn't ever have the gentle intro to calculus or linear algebra in high school, beginning in an abstract setting would be pretty tough. That might be part of the reason why calculus is (most often) presented before progression into real analysis. My first discussion on...
15. ### What is the simplest way of selecting the last N terms of a polynomial?

That only works for polynomials which can be expressed in the form (1 + x)n, though ...