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.
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...
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...
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...
^ 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.
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...
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) =...
^ 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!
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...
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...
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...
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...