superconvergence

A sequence $x_{0},x_{1},\dots$ superconverges to 0 if, when the $x_{i}$ are written in base 2, then each number $x_{i}$ starts with $2^{i}-1\approx 2^{i}$ zeroes. For example, the following sequence is superconverging to 0.

\begin{array}[]{clcl}x_{n+1}&=x_{n}^{2}&(x_{n})_{10}&(x_{n})_{2}\\ x_{0}&=&\frac{1}{2}&.1\\ x_{1}&=&\frac{1}{4}&.01\\ x_{2}&=&\frac{1}{16}&.0001\\ x_{3}&=&\frac{1}{256}&.00000001\\ x_{4}&=&\frac{1}{65536}&.0000000000000001\end{array}

In this case it is easy to see that the number of binary 0’s doubles each $x_{n}$ .

A sequence $\{x_{i}\}$ superconverges to $x$ if $\{x_{i}-x\}$ superconverges to 0, and a sequence $\{y_{i}\}$ is said to be superconvergent if there exists a $y$ to which the sequence superconverges.

Title	superconvergence
Canonical name	Superconvergence
Date of creation	2013-03-22 11:58:12
Last modified on	2013-03-22 11:58:12
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	15
Author	mathcam (2727)
Entry type	Definition
Classification	msc 41A25
Synonym	superconverge
Related topic	NewtonsMethod
Related topic	KantorovitchsTheorem
Related topic	SuperincreasingSequence