无限简单连分数的计算及其应用

提出两种无限简单连分数的求值方法.连分数首先被表示为数列的递推关系式.如果数列为收敛数列,那么无限连分数的值即为数列极限.方法一是,利用求方程的方法求解数列的极限,从而得到无限连分数的值;方法二是,先利用斐波那契数列直接求出连分数对应数列的通项表达式,进而直接取通项的极限得到连分数的值.同时,利用图像法可以直观地表示连分数的迭代求值过程.另外,基于方法一的思想,构造了对于一般函数方程的迭代格式,并指出这种迭代格式可以自然引导至微分方程中的皮卡序列方法.

Calculation and Application of Infinite Ordinary Continued Fractions

Two methods were proposed for calculating infinite ordinary continued fractions. The infinite continued fraction was converted into a recurrence relation of a sequence. If the sequence is convergent, the value of the infinite continued fraction is the limit of the sequence. For the first method, the limit was obtained by solving a corresponding equation. In the second approach, the Fibonacci sequence was used to write down the explicit expression for a general term of the sequence. Then the value of the infinite continued fraction was obtained by directly taking the limit in the expression. Besides that, the iteration process of evaluating infinite continued fractions can be intuitively shown by drawing diagrams. Moreover, based on the first approach, an iterative algorithm was developed, which can naturally lead to the Picard iteration method in solving ordinary differential equations.