A Short Proof of the Simple Continued Fraction Expansion of
نویسنده
چکیده
One of the most interesting proofs is due to Hermite; it arose as a byproduct of his proof of the transcendence of e in [5]. (See [6] for an exposition by Olds.) The purpose of this note is to present an especially short and direct variant of Hermite’s proof and to explain some of the motivation behind it. Consider any continued fraction [a0, a1, a2, . . .]. Its ith convergent is defined to be the continued fraction [a0, a1, . . . , ai]. One of the most fundamental facts about continued fractions is that the ith convergent equals pi/qi, where pi and qi can be calculated recursively using
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عنوان ژورنال:
- The American Mathematical Monthly
دوره 113 شماره
صفحات -
تاریخ انتشار 2006