For each positive integers $n$, $C_n=\{(x,y)\in \mathbb R^2:x^2+(y-n)^2=n^2\}.$ Let $$C=\cup_{n\in \mathbb N} C_n.$$ Find $\overline{C}$.
My attempt:-
$x^2+(y-n)^2=n^2$ is a circle centred at $(0,n)$ with radius $n.$
$x^2+(y-n)^2=n^2 \implies x^2+y^2-2yn+n^2=n^2\implies x^2+y^2+y^2-2yn=0 \implies...
Let $\mathbb{R}$ be the field of real numbers, and let D be a function on 2x2 matrices over $\mathbb{R}$, with values in $\mathbb{R}$, such that $D(AB) = D(A)D(B)$ for all $A, B$. Suppose also that
$$D\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} \neq D\begin{pmatrix}
1 
You are correct, the answer is 1/2.
Ask your friend this question: If $2^n$ people were flipping a fair coin $n$ times each, then the probability $p(n)$ of at least one of them getting $n$ heads in a row is
$$p(n) = 1 - \left(1-\frac{1}{2^n}\right)^{2^n}$$
Observe that
$$\lim_{n\rightarrow\infty}...
I cannot really follow the reasoning you are hinting in your question, but here's my take:
To talk about density you need a topology. Since $M_n(\mathbb{C})$, the space of complex $n\times n$ matrices is finite-dimensional, a very natural notion of convergence is entry-wise; so we can consider ...
Find the order of convergence of the sequence $\{\frac{5}{6^{(\frac{7}{3})^n}} \}. $
My attempt:-
Definition of Order of convergence
We know that $\lim_{n\to \infty}\{\frac{5}{6^{(\frac{7}{3})^n}} \}=0$
Consider
$$\frac{|p_{n+1}-p|}{|p_n-p|^{\alpha}}=\frac{\frac{5}{6^{(\frac{7}{3})^{n+1}}}}{(\fr...
