Posts Tagged ‘sequence’


March 18, 2014 Leave a comment

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

June 19, 2013 Leave a comment

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\}}.

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Nice characterization of convergence

October 6, 2011 Leave a comment

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.

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Categories: Topology Tags: , , ,

Increasing sequence of real numbers

April 17, 2010 Leave a comment

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.

Categories: Olympiad, Problem Solving Tags:

Interesting sequence

April 16, 2010 Leave a comment

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.
IMO Shortlist 2002

Property of a Sequence

September 9, 2009 1 comment

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.
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