85 lines
2.3 KiB
Markdown
85 lines
2.3 KiB
Markdown
---
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id: 5900f3ad1000cf542c50fec0
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challengeType: 5
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title: 'Problem 65: Convergents of e'
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forumTopicId: 302177
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---
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## Description
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<section id='description'>
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The square root of 2 can be written as an infinite continued fraction.
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$\sqrt{2} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + ...}}}}$
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The infinite continued fraction can be written, $\sqrt{2} = [1; (2)]$ indicates that 2 repeats <i>ad infinitum</i>. In a similar way, $\sqrt{23} = [4; (1, 3, 1, 8)]$.
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It turns out that the sequence of partial values of continued fractions for square roots provide the best rational approximations. Let us consider the convergents for $\sqrt{2}$.
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$1 + \dfrac{1}{2} = \dfrac{3}{2}\\\\
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1 + \dfrac{1}{2 + \dfrac{1}{2}} = \dfrac{7}{5}\\\\
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1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2}}} = \dfrac{17}{12}\\\\
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1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2}}}} = \dfrac{41}{29}$
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Hence the sequence of the first ten convergents for $\sqrt{2}$ are:
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$1, \dfrac{3}{2}, \dfrac{7}{5}, \dfrac{17}{12}, \dfrac{41}{29}, \dfrac{99}{70}, \dfrac{239}{169}, \dfrac{577}{408}, \dfrac{1393}{985}, \dfrac{3363}{2378}, ...$
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What is most surprising is that the important mathematical constant, $e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, ... , 1, 2k, 1, ...]$.
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The first ten terms in the sequence of convergents for <var>e</var> are:
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$2, 3, \dfrac{8}{3}, \dfrac{11}{4}, \dfrac{19}{7}, \dfrac{87}{32}, \dfrac{106}{39}, \dfrac{193}{71}, \dfrac{1264}{465}, \dfrac{1457}{536}, ...$
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The sum of digits in the numerator of the 10<sup>th</sup> convergent is $1 + 4 + 5 + 7 = 17$.
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Find the sum of digits in the numerator of the 100<sup>th</sup> convergent of the continued fraction for <var>e</var>.
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</section>
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## Instructions
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<section id='instructions'>
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</section>
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## Tests
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<section id='tests'>
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```yml
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tests:
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- text: <code>convergentsOfE()</code> should return a number.
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testString: assert(typeof convergentsOfE() === 'number');
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- text: <code>convergentsOfE()</code> should return 272.
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testString: assert.strictEqual(convergentsOfE(), 272);
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```
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</section>
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## Challenge Seed
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<section id='challengeSeed'>
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<div id='js-seed'>
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```js
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function convergentsOfE() {
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return true;
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}
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convergentsOfE();
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```
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</div>
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</section>
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## Solution
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<section id='solution'>
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```js
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// solution required
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```
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</section>
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