e和π的无理性
The Irrationality of $e$ and $π$
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中文总结 AI 辅助
研究e和π的无理性，通过数值估计连分数、利用里卡蒂方程及相关递推关系证明e的无理性，给出相关积分动机及封闭形式积分表示，还为π无理性证明提供捷径。
中文摘要 AI 辅助
与具有无限分子和分母的普通分数不同，无限连分数必定是无理数。欧拉首次通过数值估计自然对数的底数e的连分数，并利用里卡蒂方程表明如此得到的收敛分数序列pn/qn收敛于e，从而证明e是无理数。埃尔米特表明这些收敛分数的递推关系对应于某些反常积分之间的递推关系，进而证明在n趋于无穷时连分数趋于e。本文给出了相关积分的动机，并得到了所有n时pn和qn的封闭形式积分表示。有趣的是，上述结果为π无理性的标准证明提供了捷径。
英文摘要
Unlike an ordinary fraction with an infinite numerator and denominator, an infinite continued fraction must be irrational. Euler was the first to show that the base of the natural logarithm $e$ is irrational, by numerically estimating its continued fraction, and showing the infinite sequence of convergents $p_n/q_n$ so obtained converged to $e$ using the Ricatti equation. Hermite showed that the recurrence relations for these convergents correspond to the recurrence relations between certain improper integrals, so proving the continued fraction tends to $e$ in the limit of infinite $n$. Here we provide a motivation for the integrals involved and obtain closed form integral representations for $p_n$ and $q_n$ for all $n$. An interesting feature, is that the above results provide a short cut to the standard proof of the irrationality of $π$.
