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...
Does your school's math curriculum "satisfy" you?
How much rigor is in your math courses? My school has a distinct math faculty (our math program is through the math faculty, not through sciences) with a variety of math majors: combinatorics, statistics, pure math, applied math, computation...
^ Wut?? That's the result of convergence of a geometric series, it's very different.
A sequence of real numbers (x_n) converges when there exists a number L \in \mathbb{R} such that, for any \epsilon > 0, I can find a number N \in \mathbb{N} so that |x_n - L| < \epsilon whenever n > N. This is...
Mepris, I recommend either of these books: Linear Algebra by Friedberg, Insel and Spence, or Linear Algebra by Hoffman and Kunze. Both good reads. The latter is slightly advanced but it seems like that's what you're interested in.
As for partial fractions ... the coolest proof I've seen is a...
^ I think he might just mean an (ordered) set of terms from the polynomial. The order is established by listing the coefficients in descending order of their degree.
I think what you want isn't so complicated. It's as follows: if p \in \mathbb{R}\left[x\right] is a polynomial with degree n...
Proving the "proof by contradiction" method
This can get a little bit fundamental or "axiomatic", if you will. Let's say we can describe sets by prescribing a fixed property P on objects of a certain type, and claiming that a set is a collection of objects satisfying P; i.e. A = \{x : P(x)\}...
Sounds good to me too, I can contribute online if possible. Feel free to inbox me.
I have no background in analysis -- my first complex analysis course begins in January and I'll take Real Analysis I in May. Right now I'm taking my third calculus course in which we're discussing partial...
If you believe you have a good algebra (i.e. symbol manipulation) background, and understanding of basic and transcendental functions and basic geometry (polynomials, exponentials/logarithms, trigonometric functions; geometry of the circle, triangle and line), then I think Spivak is a great...
I've always thought that this was a result of a convenient choice of notation and measurement. Our units, although naturally chosen, are still human constructs. If we keep building on these constructs to develop things like calculus, then of course we will well-approximate physical phenomena --...
"1-norm" is larger than the Euclidean norm
Define, for each \vec{x} = (x_1, \ldots, x_n) \in \mathbb{R}^n, the (usual) Euclidean norm \Vert{\vec{x}}\Vert = \sqrt{\sum_{j = 1}^n x_j^2} and the 1-norm \Vert{\vec{x}}\Vert_1 = {\sum_{j = 1}^n |x_j|}.
How can we show that, for all \vec{x} \in...
The relationship is nonlinear -- you can't find a common difference or ratio between each pair of adjacent terms. The relationship is as follows: if the term in question is 1 \over n, then the following term will be 1 \over n + 1. It's not really like you can add or multiply some constant to get...
To show that f is monotonically increasing, we need to show that for any \Delta{x} > 0, f(x + \Delta{x}) > f(x) for all x in the domain; or equivalently, f(x + \Delta{x}) - f(x) > 0. An equivalent definition is that f(x_1) < f(x_2) for all x_1, x_2 in the domain of f with x_1 < x_2.
For your...
Advanced calculus, as in "calculus for the sake of calculus", is not really something science/engineering majors need to focus on. That's why applied calculus is offered at pretty much any college. But I disagree when you say that it is a required course in college -- it isn't, outside of...
^ What's "LIM"? Do you mean n is any positive integer?
I suppose the job to be done here is to find which value of k maximizes the expression in question. Let Q_n(k) be that expression. The problem is to find k_0 such that, for each n, Q_n(k_0) \geq Q_n(k) for all k -- and subsequently, to show...
It's pretty inaccurate in a rigorous mathematical context, for sure. But I bet the book you were reading wasn't really oriented for pure math, rather for engineering or science, amirite?
I see that you were referring to yourself there. But you make your point by saying that you are a mathematician, justifying that you should not write that. It's pretty clear that you're just saying "a mathematician should not write something like this". Of course I don't disagree with you in...
^ You say that like one cannot call themselves a mathematician if they write that. I'd write it if someone asks why it is written, which is exactly what's going on in this thread.
I wouldn't say it's wrong. Just not 100% mathematically precise. There is room for error (literally) in the context...
^ Not massively. Only by a small error. ;)
The appropriate amount of rigor depends on context. Engineers will use this "equation" because it helps in applications. Opticians frequently use \theta in place of \sin\theta when |\theta| is small. It's less precise for sure, but it's a useful...
By "smaller and smaller", I mean "in the limit as h approaches zero". In a lot of applications, using "tiny values" is more useful even though there is loss of precision. Of course there is always an error in difference. that's the definition of limit.
It comes directly from the definition of the derivative. Write f'(x) as follows:
f'(x) = \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h}
The limit is taken as h goes to 0, i.e. |h| is small but not zero. We can rewrite the limit as an approximation for small h:
f'(x) \approx \frac{f(x + h) -...
Check http://summer-time-nerd.tumblr.com/post/8628678169" [Broken] out (it's my blog :)). Apparently Bernoulli (one of them :p) explored this finance problem, where he encountered this limit in this theoretical situation.
Exactly. Great that you're understanding this, glad I could help. And yeah, make sure you define (or at least are clear about) what "f" is before you use it, as with any mathematical object.
^ "y" is the name of the function, "y(x)" is the particular function value when we evaluate the function at x. If you choose to call the function "f", that's fine too. I think tiny-tim's post was pretty misleading, but he made a valid point -- nowhere in your opening post did you say that (the...
Alright, then wouldn't we be interested in defining what a differential is first, so we can define operations on them?
I think in the elementary, traditional sense, "division" here doesn't really work. I've always thought \frac{\mathrm dy}{\mathrm dx} was cool-looking, but a bit notationally...
Don't you really have to define "divide" (namely, "division") in order to do that? But "divide" typically refers to an operation involving numbers -- and differentials aren't numbers.
It looks like ordinary division of numbers, but \mathrm dx and \mathrm dy are not ordinary numbers. However, we manipulate them symbolically in a way that appears like they are real numbers, for the sake of intuition. But we can do this without loss of precision! A good demonstration of it is...
^ Well, the terms in the sequence are indexed by natural numbers, so it wouldn't make sense to use another number as the index. I see that n can be a natural number while the "fixed" N doesn't have to be; but I think it might be better for teaching to think of N as "the point in the sequence...
Right. I was thinking N = \lceil{\frac{1}{\epsilon}}\rceil would work, but I didn't want to introduce the ceiling function because this is for a teaching exercise. Writing N > \frac{1}{\epsilon} - 1 is a good idea too, I was thinking more about writing N = N(\epsilon) explicitly but this works...
Call {a1, a2, a3, ...} = {an} a "convergent sequence" if
\exists L \in \mathbb{R} : \quad \forall \epsilon > 0 \quad \exists N \in \mathbb{N} : (\forall n > N \quad (n > N \implies |a_n - L| < \epsilon))
in which case we write \lim_{n \rightarrow \infty} a_n = \lim a_n = L. Of course this...
Is acos(x) the inverse cosine (arccos x), or is it a\cos x? In the latter case, I don't think there's a particular simplification for the expression -- sine takes angles as arguments, and a\cos(x) is not interpreted as an angle.
If you mean \sin(\arccos(x)), then x is an angle. Let \theta =...
^ But then to avoid being suspected of cheating, you'd have to wear full-sleeve shirts during your exams. And that sucks when your exams are seated outside during a heatwave :(
Actually, you can differentiate it everywhere except at x = 1. The function is continuous everywhere, just not differentiable at that particular number.