MathDB

6

Part of 2000 IMC

Problems(2)

hard limit

Source: IMC 2000 day 1 problem 6

10/29/2005
Let f:R→]0,+∞[f: \mathbb{R}\rightarrow ]0,+\infty[ be an increasing differentiable function with lim⁡x→+∞f(x)=+∞\lim_{x\rightarrow+\infty}f(x)=+\infty and f′f' is bounded, and let F(x)=∫0xf(t)dtF(x)=\int^x_0 f(t) dt. Define the sequence (an)(a_n) recursively by a0=1,an+1=an+1f(an)a_0=1,a_{n+1}=a_n+\frac1{f(a_n)} Define the sequence (bn)(b_n) by bn=F−1(n)b_n=F^{-1}(n). Prove that lim⁡x→+∞(an−bn)=0\lim_{x\rightarrow+\infty}(a_n-b_n)=0.
functionlimitintegrationreal analysisreal analysis unsolved
Nilpotent Matrices

Source: IMC 2000 Day 2 Problem 6

10/27/2020
Let AA be a real n×nn\times n Matrix and define eA=∑k=0∞Akk!e^{A}=\sum_{k=0}^{\infty} \frac{A^{k}}{k!} Prove or disprove that for any real polynomial P(x)P(x) and any real matrices A,BA,B, P(eAB)P(e^{AB}) is nilpotent if and only if P(eBA)P(e^{BA}) is nilpotent.
linear algebramatrixalgebrapolynomial