Competition problem — Number theory

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Number theory Difficulty 5.5 AIME, harder Find the answer

4. The infinite simple continued fraction expansion of the number $e$ is
$e=[2 ; 1,2,1,1,4,1,1,6,1,1,8, \ldots]$
a) Find the first eight convergents of the continued fraction of $e$ .
b) Find the best rational approximation to $e$ having a denominator less than 100.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

4. a) $\frac{2}{1}, \frac{3}{1}, \frac{8}{3}, \frac{11}{4}, \frac{19}{7}, \frac{87}{32}, \frac{106}{39}, \frac{193}{71}$
b) $\frac{193}{71}$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.