Problem of the Week

Problem 4 (due Monday, March 31 )

A sequence $a_1,a_2,\ldots$ of real numbers has the following properties:

(i) $|a_1+a_2+\ldots +a_k|\leq 1$ for every $k$ ;

(ii) $|a_k-a_{k-1}|\leq 1/k$ for every $k\geq 2$ .

Suppose that $\displaystyle |a_k|\geq \frac{c}{\sqrt{k}}$ for infinitely many $k$ . Prove that $c\leq \sqrt{2}$ .

Overview

Every other Monday (starting 01/27/25), we will post a problem to engage our mathematical community in the problem solving activity and to enjoy mathematics outside of the classroom. Students (both undergraduate and graduate) are particularly encouraged to participate as there is no better way to practice math than working on challenging problems. If you have a solution and want to be a part of it, e-mail your solution to Marcin Mazur (mazur@math.binghamton.edu) by the due date. We will post our solutions as well as novel solutions from the participants and record the names of those who've got the most number of solutions throughout each semester.

When you submit your solutions, please provide a detailed reasoning rather than just an answer. Also, please include some short info about yourself for our records.

Previous Problems and Solutions

Problem 3 Solved by Ashton Keith (Purdue University), Josiah Moltz, and Dr Mathew Wolak.

Problem 2 Solved by Josiah Moltz and Dr Mathew Wolak.

Problem 1 Solution submitted by David Biddle, ChatGPT via Prof. Vladislav Kargin, Colin McCann jointly with Michael Mancuso, Maxwell Meyers, Ethan Tong.