147 lines
1.7 KiB
Markdown
147 lines
1.7 KiB
Markdown
---
|
|
id: 5900f3ad1000cf542c50fec0
|
|
challengeType: 5
|
|
title: 'Problem 65: Convergents of e'
|
|
forumTopicId: 302177
|
|
---
|
|
|
|
## Description
|
|
<section id='description'>
|
|
The square root of 2 can be written as an infinite continued fraction.
|
|
|
|
√2 = 1 +
|
|
1
|
|
|
|
2 +
|
|
1
|
|
|
|
|
|
2 +
|
|
1
|
|
|
|
|
|
|
|
2 +
|
|
1
|
|
|
|
|
|
|
|
|
|
2 + ...
|
|
|
|
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)].
|
|
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.
|
|
|
|
|
|
1 +
|
|
1
|
|
= 3/2
|
|
|
|
2
|
|
|
|
1 +
|
|
1
|
|
= 7/5
|
|
|
|
2 +
|
|
1
|
|
|
|
|
|
2
|
|
|
|
1 +
|
|
1
|
|
= 17/12
|
|
|
|
2 +
|
|
1
|
|
|
|
|
|
|
|
2 +
|
|
1
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
1 +
|
|
1
|
|
= 41/29
|
|
|
|
2 +
|
|
1
|
|
|
|
|
|
2 +
|
|
1
|
|
|
|
|
|
|
|
|
|
2 +
|
|
1
|
|
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
Hence the sequence of the first ten convergents for √2 are:
|
|
1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, ...
|
|
What is most surprising is that the important mathematical constant,e = [2; 1,2,1, 1,4,1, 1,6,1 , ... , 1,2k,1, ...].
|
|
The first ten terms in the sequence of convergents for e are:
|
|
2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, ...
|
|
The sum of digits in the numerator of the 10th convergent is 1+4+5+7=17.
|
|
Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e.
|
|
</section>
|
|
|
|
## Instructions
|
|
<section id='instructions'>
|
|
|
|
</section>
|
|
|
|
## Tests
|
|
<section id='tests'>
|
|
|
|
```yml
|
|
tests:
|
|
- text: <code>euler65()</code> should return 272.
|
|
testString: assert.strictEqual(euler65(), 272);
|
|
|
|
```
|
|
|
|
</section>
|
|
|
|
## Challenge Seed
|
|
<section id='challengeSeed'>
|
|
|
|
<div id='js-seed'>
|
|
|
|
```js
|
|
function euler65() {
|
|
// Good luck!
|
|
return true;
|
|
}
|
|
|
|
euler65();
|
|
```
|
|
|
|
</div>
|
|
|
|
|
|
|
|
</section>
|
|
|
|
## Solution
|
|
<section id='solution'>
|
|
|
|
```js
|
|
// solution required
|
|
```
|
|
|
|
</section>
|