MathDB

6

Part of 2003 IMC

Problems(2)

polynomial

Source: IMC-2003

5/5/2004
Let p=∑k=0nakXk∈R[X] p=\sum\limits_{k=0}^n a_kX^k\in R[X] a polynomial such that all his roots lie in the half plane {z∈C∣Re(z)<0}. \{z\in C| Re(z)<0 \}. Prove that akak+3<ak+1ak+2, a_ka_{k+3}<a_{k+1}a_{k+2}, for every k=0,1,2...,n-3.
algebrapolynomialreal analysisreal analysis unsolved
IMC 2003 Problem 12

Source: IMC 2003 Day 2 Problem 6

11/2/2020
Let (an)(a_{n}) be the sequence defined by a0=1,an+1=∑k=0nakn−k+2a_{0}=1,a_{n+1}=\sum_{k=0}^{n}\dfrac{a_k}{n-k+2}. Find the limit lim⁡n→∞∑k=0nak2k,\lim_{n \rightarrow \infty} \sum_{k=0}^{n}\dfrac{a_{k}}{2^{k}}, if it exists.
limitConvergencereal analysis