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AN EXACT FORMULA FOR THE HARMONIC CONTINUED FRACTION
Bulletin of the Australian Mathematical Society ( IF 0.7 ) Pub Date : 2020-06-10 , DOI: 10.1017/s0004972720000544
MARTIN BUNDER , PETER NICKOLAS , JOSEPH TONIEN

For a positive real number $t$, define the harmonic continued fraction $$\begin{eqnarray}\text{HCF}(t)=\biggl[\frac{t}{1},\frac{t}{2},\frac{t}{3},\ldots \biggr].\end{eqnarray}$$ We prove that $$\begin{eqnarray}\text{HCF}(t)=\frac{1}{1-2t(\frac{1}{t+2}-\frac{1}{t+4}+\frac{1}{t+6}-\cdots \,)}.\end{eqnarray}$$

中文翻译:

谐波连分数的精确公式

对于正实数$t$, 定义调和连分数$$\begin{eqnarray}\text{HCF}(t)=\biggl[\frac{t}{1},\frac{t}{2},\frac{t}{3},\ldots \biggr ].\end{eqnarray}$$我们证明$$\begin{eqnarray}\text{HCF}(t)=\frac{1}{1-2t(\frac{1}{t+2}-\frac{1}{t+4}+\frac {1}{t+6}-\cdots \,)}.\end{eqnarray}$$
更新日期:2020-06-10
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