Let $f: U \subset \mathbb{R}^2 \to \mathbb{R}$ where $U$ is open and connected. Suppose $f_x, f_y = 0$ on $U$. Can we conclude from here that $f$ must me constant ?

Question: If a,b are 2 positive , co-prime integers such that $$\lim _{n \rightarrow \infty}(\frac{^{3n}C_n}{^{2n}C_n})^\frac{1}{n}=\frac{a}{b}$$Then a+b?? I tried to break down the limit to: $$\frac{[(3n)(3n-1)\dot{}\dot{}\dot{})^\frac{1}{n}][n(n-1)(n-2)\dot{}\dot{}\dot{})^\frac{1}{n}]}{[2n(2n-1)(2n-2)\dot{}\dot{}\dot{}]^\frac{2}{n}}$$ but I’m lost ahead of it. Answer given in my text books is 43. Please guide me

Consider the function $f(x)=1-x-(1-\frac{a}{n}x)^n$ where $a$ and $n$ are both constants such that $\frac{a}{n}\in[0,1]$ ( it represents some probability indeed). One obvious solution is $x=0$. I want to prove that such a function has a real solution between $(0,1]$ and this solution has the form that dependening on $a$ and independent of $n$. The only […]

I dont know how to calculate these two series: $$\begin{align} & \sum\limits_{n=1}^{\infty }{\frac{n+3}{{{n}^{3}}+\ln n}} \\ & \sum\limits_{n=1}^{\infty }{\left( 1-\cos \frac{\pi }{n} \right)} \\ \end{align}$$

I found a series of articles (in Portuguese) by a Brazilian mathematician named Ludenir Santos, where presents a series of iterative methods, he said new, to extract nth roots of any complex number different from zero (and hence for real numbers), with any desired degree of accuracy. In his first articles, he has two different methods […]

Let $g$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}g(t)\leq C\int_{r}^{\infty}g(t)\frac{dt}{t}, $$ where positive constant $C$ does not depend on $r>0$.

Claim Let $P_n: \Bbb R \mapsto \Bbb R $, $n \in \Bbb N\cup\{0\}$ $P_n(x):={1\over{2^n{n!}}}{d^n\over dx^n}[(x^2-1)^n]$, then $P_n$ has n distinct roots in ]-1, 1[ Then show that below equation is fulfilled. $(1-x^2)P_n^{”}(x) – 2xP_n^{‘}(x) + n(n+1)P_n^{}(x) = 0$ Question I had tried brute way, but it doesn’t conclude to 0. and also induction doesn’t work […]

Let $X, \mathcal{P}(X) = M$ be a $\sigma$-algebra. Let $f: X \rightarrow [0,\infty]$ be a function. Define $\mu : M \rightarrow [0,\infty]$ by $\mu(E) = \sum_{x \in E} f(x)$, which Folland defines as $$\sup_{\substack{F \subset E \\ F \text{ finite}}} \sum_{x \in F} f(x).$$ Prove that $\mu$ is a measure. It’s easy to show that […]

Show that if $f$ is a continuous function on the interval $[a , b]$ and if $\int_a^b f(x) p(x)dx = 0$ for every polynomial $p$ then $f$ must be the zero function. Attempt: By the Weierstrass Approximation Theorem, we know there is a sequence $\{p_j\}$ such that $p_j \rightarrow f$ uniformly on $[a,b]$. (The Theorem […]

I’ve come across a lot of websites that were giving a proof of the density of $\mathbb{Q}$ in $\mathbb{R}$, but it seems to me that none of them was using only what we had already covered in my lecture, so I tried to prove it by myself (with what we had already seen). The thing […]

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