Thanx. But I think this approach is beyond the scope of my course. I was specifically told to look at the taylor expansion of ln(2). Is there a way to do it that way?
Homework Statement
As part of a problem I have to show that lim_{n\to\infty}\sum_{i=\frac{n}{2}}^{n}\frac{1}{i}=ln(2)
Homework Equations
Taylor expansion of ln(2): \sum_{i=1}^{\infty}\frac{(-1)^{k+1}}{k}
The Attempt at a Solution
ln(2) can be written as: ln(2) =...
I have to prove that the cardinality of the set of infinite sequences of real numbers is equal to the cardinality of the set of real numbers. So:
A := |\mathbb{R}^\mathbb{N}|=|\mathbb{R}| =: B
My plan was to define 2 injective maps, 1 from A to B, and 1 from B to A.
B <= A is trivial, just...
This is not really a homework question, but I've come across this while preparing for a test
Homework Statement
Let f:U \subseteq R^n -> R^m be a function which is differentiable at a \in U, and u \in R^n
It is then stated that it is clear that:
lim_{t \to 0}...
With homogeneous I just ment the differential equation without the constant part.
But your approach won't work because u'(t)=u_1'(t)+u_2'(t)=-c(a+b)u_1(t)+cu_1^2(t)-c(a+b)u_2(t)+cab=-c(a+b)u(t)+cu_1^2(t)+cab
Or am I seeing it wrong?
Homework Statement
Find a solution for:
u'(t)=c*u(t)^2-c*(a+b)*u(t)+c*a*b
The Attempt at a Solution
I've found the solution for the homogeneous equation:
u_0(t)=(\frac{1}{a+b}+d*e^{c(a+b)t)})^{-1}
Where c is a random constant.
Now I've tried the solution u(t)=x(t)*u_0(t), when I fill...
Hi there,
Im looking for a theorem that relies on the axiom of choice, but is used in applied mathematics (economics, physics, biology, whatever). In other words a mathematical theory we use to say something about the real world.
This is because I'm wondering if discarding the axiom of...
Homework Statement
Given an infinite long (in the z-direction), with width 2b in the y-direction and no thickness in the x-direction. There is a current with density J in the z-direction.
Calculate the magnetic field on the y-axis if |y| > b.
Homework Equations
Amperes law: ∫B*dS = μ0*I...
So if I say something like: 7x^2+2y^2 \leq 6 \Rightarrow y^2 \leq 3 \lt 4 \Rightarrow y \lt 2
and 7x^2+2y^2 \leq 6 \Rightarrow x^2 \leq \frac{6}{7} \lt 1 \Rightarrow x \lt 1
and z \leq x^2y+5y^3 \Rightarrow z \lt (2+5*8)=42
So choose M = 4+1+42^2 = 1769. And this M will do.
Homework Statement
Show that D = { (x,y,z) \in \mathbb{R}^{3} | 7x^2+2y^2 \leq 6, x^3+y \leq z \leq x^2y+5y^3} is bounded.
Homework Equations
Definition of bounded:D \subseteq \mathbb{R}^{n} is called bounded if there exists a M > 0 such that D \subseteq \{x \in \mathbb{R}^{n} | ||x|| \leq...
X would be it's power set, and thus there would be a bijection between them. However I've proven there doesn't exist a bijection. In other words, X and it's power set have the same cardinality (if X exists), but I've proven the power set is bigger than X, so there is a contradiction and thus X...
Homework Statement
Prove a set which contains all of it's subsets doesn't exist.
The Attempt at a Solution
Suppose such a set P exists. P := {x | x \in \wp(x)}.
P \in \wp(x), so P \in P.
This seems like a paradox to me, so all I have to prove is that a set can't contain itself. But how...