Thinking Cap

Problem of the Month


November 2019

Breaking into small pieces

Show that for every natural number \(k>1\) there exist natural numbers \(n\) and \(m\) with \(n>m\) such that \[ \frac{1}{k} = \frac{1}{m(m+1)} + \frac{1}{(m+1)(m+2)} + \cdots + \frac{1}{n(n+1)} \]