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Posts Tagged ‘limit’

IMC 2014 Day 1 Problem 2

July 31, 2014 2 comments

Consider the following sequence

\displaystyle (a_n)_{n=1}^\infty = (1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,...)

Find all pairs {(\alpha,\beta)} of positive real numbers such that

\displaystyle \lim_{n \rightarrow \infty} \frac{\sum_{k=1}^n a_k}{n^\alpha}=\beta.

IMC 2014 Day 1 Problem 2

Categories: Analysis, Olympiad Tags: , ,

SEEMOUS 2014

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)}

Asymptotic characterization in terms of sequence limits

December 17, 2013 Leave a comment

Suppose {f:(0,\infty) \rightarrow \Bbb{R}} is a continuous function such that for every {x>0} we have

\displaystyle \lim_{n \rightarrow \infty} f(nx)=0.

Prove that {\lim\limits_{x \rightarrow \infty} f(x)=0}.

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SEEMOUS 2012 Problem 4

March 8, 2012 Leave a comment

a) Compute \displaystyle \lim_{n \to \infty} n \int_0^1 \left(\frac{1-x}{1+x} \right)^n dx.

b) Let k \geq 1 be an integer. Compute

\displaystyle\lim_{n \to \infty}n^{k+1}\int_0^1 \left( \frac{1-x}{1+x}\right)^n x^k dx.

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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: , , ,

Convex function & limit

January 6, 2010 Leave a comment

Suppose f: [0,\infty) \to \mathbb{R} is convex and differentiable with \lim\limits_{x\to \infty} \frac{f(x)}{x}=\ell \in \mathbb{R}. Prove that \lim\limits_{x \to \infty} f^\prime (x)=\ell . Read more…

Categories: Analysis, Problem Solving Tags: ,

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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