Articles of real analysis

Characterizing the continuum using only the notion of midpoint

Is it possible to categorically characterize the continuum using only the notion of midpoint? I assume some kind of Dedekind-cut-axiom will be necessary, but I don’t know how to define order (see my related question).

if derivative is zero, then function is constant

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 ?

How to find this limit with following constraints?

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

Prove the function $f(x)=1-x-(1-\frac{a}{n}x)^n$ has a real solution between $(0,1]$

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 […]

Two simple series

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

An apparently new method to compute the $n$th root of any complex number

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 […]

An inequality involving supremum and integral

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 $r>0$.

For Legendre's Equation $P_n(x)$: $(1-x^2)P_n^{''}(x) – 2xP_n^{'}(x) + n(n+1)P_n^{}(x) = 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 […]

Generalization of counting measure is a measure

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 […]

If $f$ is continuous on $$ and $\int_a^b f(x) p(x)dx = 0$ then $f = 0$

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 […]