### Archive

Posts Tagged ‘series’

## IMC 2016 – Day 2 – Problem 6

July 28, 2016 1 comment

Problem 6. Let ${(x_1,x_2,...)}$ be a sequence of positive real numbers satisfying ${\displaystyle \sum_{n=1}^\infty \frac{x_n}{2n-1}=1}$. Prove that

$\displaystyle \sum_{k=1}^\infty \sum_{n=1}^k \frac{x_n}{k^2} \leq 2.$

## Vojtech Jarnik Competition 2015 – Problems Category 2

Problem 1. Let ${A}$ and ${B}$ be two ${3 \times 3}$ matrices with real entries. Prove that

$\displaystyle A - (A^{-1}+(B^{-1}-A)^{-1})^{-1} = ABA,$

provided all the inverses appearing on the left-hand side of the equality exist.

Problem 2. Determine all pairs ${(n,m)}$ of positive integers satisfying the equation

$\displaystyle 5^n = 6m^2+1.$

Problem 3. Determine the set of real values ${x}$ for which the following series converges, and find its sum:

$\displaystyle \sum_{n=1}^\infty \left( \sum_{k_i \geq 0, k_1+2k_2+...+nk_n = n} \frac{(k_1+...+k_n)!}{k_1!...k_n!} x^{k_1+...+k_n}\right).$

Problem 4. Find all continuously differentiable functions ${f : \Bbb{R} \rightarrow \Bbb{R}}$, such that for every ${a \geq 0}$ the following relation holds:

$\displaystyle \int_{D(a)} xf\left( \frac{ay}{\sqrt{x^2+y^2}}\right) dxdydz = \frac{\pi a^3}{8}(f(a)+\sin a -1),$

where ${D(a) = \left\{ (x,y,z) : x^2+y^2+z^2 \leq a^2,\ |y| \leq \frac{x}{\sqrt{3}}\right\}}$

## Divergence

Prove that if $c<2$ then the series $\displaystyle \sum\limits_{n \geq 1} \frac{1}{n^{c+\sin n}}$ diverges.

Categories: Analysis, Problem Solving Tags:

## Every expansion has a 0

Suppose $f$ is entire such that for each $x_0 \in \mathbb{C}$ at least one of the coefficients of the expansion $f=\sum\limits_{n=0}^\infty c_n (z-z_0)^n$ is equal to 0. Prove that $f$ is a polynomial.
Let $F(z)=\sum\limits_{n=1}^\infty d(n) z^n$ where $d(n)$ denotes the number of divisors of $n$. Calculate the radius of convergence of this series and prove that $F(z)=\sum\limits_{n=1}^\infty \frac{z^n}{1-z^n}$. Furthermore, prove that $F(r)\geq \frac{1}{1-r}\log \frac{1}{1-r}$ for $r \in (0,1)$.
Suppose $(a_n)$ is a non-increasing sequence of positive real numbers and $\varepsilon_i \in \{\pm 1\},\ \forall i \in \mathbb{N}$ such that $\sum\limits_{i=1}^\infty \varepsilon_i a_i$ is convergent.
Prove that $\lim\limits_{n\to \infty}(\varepsilon_1+\varepsilon_2+...+\varepsilon_n) a_n=0$.