144 lines
1.7 KiB
Markdown
144 lines
1.7 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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---
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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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√2 = 1 +
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1
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2 +
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1
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2 +
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1
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2 +
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1
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2 + ...
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The infinite continued fraction can be written, √2 = [1;(2)], (2) indicates that 2 repeats ad infinitum. In a similar way, √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 √2.
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1 +
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1
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= 3/2
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2
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1 +
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1
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= 7/5
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2 +
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1
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2
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1 +
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1
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= 17/12
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2 +
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1
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2 +
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1
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2
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1 +
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1
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= 41/29
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2 +
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1
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2 +
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1
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2 +
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1
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2
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Hence the sequence of the first ten convergents for √2 are:
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1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 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 e are:
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2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, ...
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The sum of digits in the numerator of the 10th convergent is 1+4+5+7=17.
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Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e.
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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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- text: <code>euler65()</code> should return 272.
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testString: 'assert.strictEqual(euler65(), 272, "<code>euler65()</code> should return 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 euler65() {
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// Good luck!
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return true;
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}
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euler65();
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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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