A way to remove duplication from ∑^N_i=1 1/x_i

The main purpose of this paper is to show that we can rewrite a sum of unit fractions ∑^N_{i = 1} 1/x_i< 1 in which 0<x_i≤x_{i+ 1} to the form ∑^N_{i = 1} 1/x′_i in which 0 <x′_i<x′_{i+ 1}. In other words, it is always possible to remove duplication from ∑^N_{i = 1} 1/x_i, without changing its length. Moreover, using this rewriting process, we get a new algorithm for the expansion of Egyptian fractions. We state this algorithm and show some numerical results.