MathDB
Infinite product

Source: IMC 1997 day 1 problem 4

October 1, 2005
real analysisreal analysis unsolved

Problem Statement

Let α\alpha be a real number, 1<α<21<\alpha<2. (a) Show that α\alpha can uniquely be represented as the infinte product α=(1+1n1)(1+1n2)⋯ \alpha = \left(1+\dfrac1{n_1}\right)\left(1+\dfrac1{n_2}\right)\cdots with nin_i positive integers satisfying ni2≤ni+1n_i^2\le n_{i+1}. (b) Show that α∈Q\alpha\in\mathbb{Q} iff from some kk onwards we have nk+1=nk2n_{k+1}=n_k^2.
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