# superconvergence

A sequence ${x}_{0},{x}_{1},\mathrm{\dots}$ *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.

$$\begin{array}{cccc}\hfill {x}_{n+1}\hfill & ={x}_{n}^{2}\hfill & \hfill {({x}_{n})}_{10}\hfill & {({x}_{n})}_{2}\hfill \\ \hfill {x}_{0}\hfill & =\hfill & \hfill \frac{1}{2}\hfill & .1\hfill \\ \hfill {x}_{1}\hfill & =\hfill & \hfill \frac{1}{4}\hfill & .01\hfill \\ \hfill {x}_{2}\hfill & =\hfill & \hfill \frac{1}{16}\hfill & .0001\hfill \\ \hfill {x}_{3}\hfill & =\hfill & \hfill \frac{1}{256}\hfill & .00000001\hfill \\ \hfill {x}_{4}\hfill & =\hfill & \hfill \frac{1}{65536}\hfill & .0000000000000001\hfill \end{array}$$ |

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 |