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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.

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Vojtech Jarnik Competition 2015 – Problems Category 2

March 27, 2015 Leave a comment

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

March 19, 2010 Leave a comment

Prove that if c<2 then the series \displaystyle \sum\limits_{n \geq 1} \frac{1}{n^{c+\sin n}} diverges.
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Every expansion has a 0

October 13, 2009 Leave a comment

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.
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Power series & number theory

October 13, 2009 Leave a comment

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).

Pseudo-alternate series

October 1, 2009 Leave a comment

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.

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