### Archive

Posts Tagged ‘limit’

## IMC 2014 Day 1 Problem 2

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

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

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

## SEEMOUS 2012 Problem 4

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

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

## 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: , , ,

## Convex function & limit

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…
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.