Engel连分数部分总和的极限定理的研究

Engel连分数部分总和的极限定理的研究(2500字)
摘要：对任意的x ∈ [0,1)，令 为x的 Engel 连分数展式.研究了Engel连分数展式的度量性质，与普通连分数一样,证明了部分商的增长性满足“0-1”率.通过构造一族恰当的集合,得到了部分商增长速度的上下极限.对x的正规化部分和 ，得到 不满足强大数定律。

关键词: Engel连分数；部分和；大数定律；“0-1”率

Limit theorems for the partial sum in Engel continued fractions
Abstract: For any x ∈ [0,1), let   be the Engel continued fraction  expansion.Similar to reg ular continued fr action, we show the grow th rate in the partial quotients satisfies the so- called 0- 1 law. Moreover, accor ding to constructing a family of proper sets, we
obtain the upper and low er limits of the speed of growth rate in the partial quotients. For the normalized partial sumt , we prove that the partial sum  can not satisfy the strong law of large numbers.
Keywords: Engel continued fractions; partial sum; strong large number law; "0-1"law

目  录
摘 要   ………………………………………………………………………………4
关键词     ………………………………………………………………………………4
1 前言     ………………………………………………………………………………5
2 Engel连分数基本性质 ……………………………………………………………6
2.1 Engel连分数基本性质的证明 …………………………………………………6
3 Engel连分数的极限定理 …………………………………………………………10
3.1 Engel连分数的极限定理的证明 ………………………………………………10
4结束语  …………………………………………………………………………… 11
参考文献    ……………………………………………………………………………12
致  谢    …………………………………………………………………………………13