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Posts Tagged ‘sequence’

SEEMOUS 2014

Problem 1. Let ${n}$ be a nonzero natural number and ${f:\Bbb{R} \rightarrow \Bbb{R}\setminus \{0\}}$ be a function such that ${f(2014)=1-f(2013)}$. Let ${x_1,..,x_n}$ be distinct real numbers. If

$\displaystyle \left| \begin{matrix} 1+f(x_1)& f(x_2)&f(x_3) & \cdots & f(x_n) \\ f(x_1) & 1+f(x_2) & f(x_3) & \cdots & f(x_n)\\ f(x_1) & f(x_2) &1+f(x_3) & \cdots & f(x_n) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ f(x_1)& f(x_2) & f(x_3) & \cdots & 1+f(x_n) \end{matrix} \right|=0$

prove that ${f}$ is not continuous.

Problem 2. Consider the sequence ${(x_n)}$ given by

$\displaystyle x_1=2,\ \ x_{n+1}= \frac{x_n+1+\sqrt{x_n^2+2x_n+5}}{2},\ n \geq 2.$

Prove that the sequence ${y_n = \displaystyle \sum_{k=1}^n \frac{1}{x_k^2-1} ,\ n \geq 1}$ is convergent and find its limit.

Problem 3. Let ${A \in \mathcal{M}_n (\Bbb{C})}$ and ${a \in \Bbb{C},\ a \neq 0}$ such that ${A-A^* =2aI_n}$, where ${A^* = (\overline A)^t}$ and ${\overline A}$ is the conjugate matrix of ${A}$.

(a) Show that ${|\det(A)| \geq |a|^n}$.

(b) Show that if ${|\det(A)|=|a|^n}$ then ${A=aI_n}$.

Problem 4. a) Prove that ${\displaystyle \lim_{n \rightarrow \infty} n \int_0^n \frac{\arctan(x/n)}{x(x^2+1)}dx=\frac{\pi}{2}}$.

b) Find the limit ${\displaystyle \lim_{n \rightarrow \infty} n\left(n \int_0^n \frac{\arctan(x/n)}{x(x^2+1)}dx-\frac{\pi}{2} \right)}$

Agregation 2013 – Analysis – Part 3

Part III: Muntz spaces and the Clarkson-Edros Theorem

Recall that for every ${\lambda \in \Bbb{N}}$ we define ${nu_\lambda(t)=t^\lambda,\ t \in [0,1]}$ and that ${\Lambda=(\lambda_n)_{n \in \Bbb{N}}}$ is a strictly increasing sequence of positive integers.

1. Suppose that ${\lambda_0=0}$ and ${\displaystyle \sum_{n \geq 1}\frac{1}{\lambda_n} =\infty}$. Let ${k \in \Bbb{N}\setminus \Lambda}$. Define ${Q_0=\nu_k}$ and by recurrence for ${n \in \Bbb{N}}$ define ${Q_{n+1}}$ by

$\displaystyle Q_{n+1}(x)=(\lambda_{n+1}x^{\lambda_{n+1}}\int_x^1 Q_n(t)t^{-1-\lambda_{n+1}}dt.$

(a) Calculate ${Q_1}$ and prove that ${\|Q_1\|_\infty \leq \displaystyle \left|1-\frac{k}{\lambda_1}\right|.}$

(b) Prove that for every ${n \geq 1}$ ${Q_n-\nu_k}$ is a linear combination of ${\nu_{\lambda_1},\nu_{\lambda_2},..,\nu_{\lambda_n}}$.

(c) Prove that for every ${n \geq 1}$ we have ${\displaystyle\|Q_n\|_\infty \leq \prod_{i=1}^n \left|1-\frac{k}{\lambda_j}\right|}$.

(d) Deduce that ${\nu_k \in \overline{M_\Lambda}}$.

(e) Conclude that ${C([0,1])=\overline{M_\Lambda}}$.

From here on suppose that ${\lambda_0}$ is arbitrary and the series ${\displaystyle \sum_{n \geq 1}\frac{1}{\lambda_n}}$ converges. For ${p \in \Bbb{N}}$ denote ${\rho_p(\Lambda)=\sum_{\lambda_n>p} \frac{1}{\lambda_n}}$. For ${s \in \Bbb{N}}$ denote ${N_s(\Lambda)}$ the cardinal of the set ${\{n \in \Bbb{N} | \lambda_n\leq s\}}$.

Nice characterization of convergence

Suppose $X$ is a topological space, and consider the sequence $(x_n)$ with the following property:

• every subsequence $(x_{n_k})$ has a subsequence converging to $x$.

Then $x_n \to x$.

Categories: Topology Tags: , , ,

Increasing sequence of real numbers

Define the sequence $(a_n)_n$ in the following way: $a_0$ is a positive integer and
$a_{n+1}=\begin{cases} \frac{a_n}{5} & 5 | a_n \\ \lfloor \sqrt{5}a_n\rfloor & 5\nmid a_n \end{cases}$
Prove that from a certain moment, the sequence is increasing.

Interesting sequence

Consider $x_1,x_2,...$ an infinite sequence such that $\displaystyle | x_i-x_j| \geq \frac{1}{i+j}$ for any $i,j$ positive integers. Prove that if $x_n \in [0,c]$ for all $n$, then $c \geq 1$.
We have bounded a sequence $(x_n)$ of real numbers such that $\lim\limits_{n\to \infty} (x_{n+1}-x_n)=0$. Prove that the set of accumulation points of such a sequence is a closed interval.