AC::MJ LEE
Simple integral Compute $\int_0^1 \frac{x^k -1}{\ln x} dx$ Proof. Define a function $f : \mathbb{R}\to \mathbb{R}\cup \{ \pm \infty \}$ by $f(k)=\int_0^1 \frac{x^k -1}{ \ln x} dx$. Since $f(0)=0-1$, $f'(k)=\int_0^1 x^k dx =\frac{1}{k+1}$. This implies that for $k>-1$, $..
Determinant of a random 0-1 matrix Let $n$ be a fixed positive integer and let $p\in(0,1)$. Let $D_n$ be the determinant of a random $n \times n$ 0-1 matrix whose entries are independent identical random variables, each of which is 1 with the probability $p$ and 0 with the probability $1-p$. Find the expected value and variance of $D_n$. Proof. If $x$ is such a random variable, $E(x)=E(x^2)=p$. ..
the Inverse of an Upper Triangular Matrix Let $A=(a_{ij})$ be an $n \times n$ upper triangular matrix such that $$a_{ij} = \binom{n-i+1}{j-i}$$ for all $i \leq j$. Find the inverse matrix of $A$. Lemma. For $i < n$, $$\sum_{k=i}^n (-1)^{n-k}\binom{k}{i} \binom{n}{k} = 0$$ Proof. Let $B=(b_{ij})$ be an $n \times n$ upper triangular matrix such that $$b_{ij} = (-1)^{j-i} \binom{n-i+1}{j-i}$$ for a..
A limit of a sequence involving a square root Let $a_0 = 3$ and $a_n = a_{n-1} + \sqrt{{a_{n-1}}^2 + 3}$ for all $n \geq 1$. Determine $\lim_{n\to\infty} \frac{a_n}{2^n}$. Proof. Note that $\cot \theta = \cot 2\theta + \sqrt{\cot^2 2\theta + 1}$ for $0 < \theta < \pi / 2$. Since $a_n $ is determined by $a_0 = 3 = \sqrt{3} \cot {\pi / 6}$, we get $$a_n = \sqrt{3} \cot (\frac{\pi}{6\times 2^n})$$ ..
Diagonal Let $r_1$, $r_2$, $r_3$,... be a sequence of all rational numbers in $(0,1)$ except finitely many numbers. Let $r_j=0.a_{j,1} a_{j,2} a_{j,3} \cdots$ be a decimal representation of $r_j$. (For instance, if $r_j = \frac{1}{3} = 0.3333\cdots$, then $a_1,k = 3$ for any $k$. Prove that the number $0.a_{1,1} a_{2,2} a_{3,3} a_{4,4} \cdots $ given by the main diagonal cannot be a rational num..
A finite ring Prove that if a finite ring has two elements $x$ and $y$ such that $xyy=y$, then $yxy=y$. Proof. Consider a subset $\{y^k | k\in \mathbb{N} \}$. Since the ring is finite, the subset is also finite, so there must exist $n$,$m\in \mathbb{N}$ such that $y^n=y^{n+m}$. If $n>1$, $$y^{n-1}=xy^{n}=xy^{n+m}=y^{n-1+m}$$ By repeating this, we get $y=y^{m+1}$. Therefore, $$yxy=yxy^{m+1}=yy^m=..
Equation with integration Determine all continuous functions $f:(0, \infty) \to (0, \infty)$ such that $\int_{t}^{t^3} f(x) dx =2 \int_{1}^{t} f(x) dx$ for all $t > 0$. Proof. It's easy to check that $f(t)=k/t$ satisfies the equation for any constant $k>0$. The purpose is to show that these are all solutions of the equation. Since $f$ is continuous, by Fundamental Theorem of Calculus, we can tak..