A generalization of the Archimedean double sequence
The authors discuss a double sequence process given by the equations \(a_{n+1}=M(a_ n,b_ n)\quad and\quad b_{n+1}=M'(a_{n+1},b_ n)\) where \(a_ 0,b_ 0>0\) and the means M and M' are continuous mappings from \({\mathbb{R}}^+\times {\mathbb{R}}^+\to {\mathbb{R}}^+\) satisfying (i) \(a\leq b\Rightarrow a\leq M(a,b)\leq b\), (ii) \(M(a,b)=M(b,a)\) and (iii) \(a=M(a,b)\Rightarrow a=b\). For given \(a_ 0,b_ 0>0\) the convergence of \((a_ n)\) and \((b_ n)\) to a common limit, say \(\alpha\), follows easily. This is a generalization of the process attributed to Archimedes for estimating \(\pi\), when \(a_ 0=3\sqrt{3}\), \(b_ 0=3\sqrt{3}/2\) and M and M' are the harmonic and geometric means respectively. The main object of the paper is to show that, for all M and M', the errors of \(a_ n- \alpha\) and \(b_ n-\alpha\) tend to zero asymptotically like \(1/4^ n\).
scientific article; zbMATH DE number 3884695