---
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.

$\sqrt{2} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + ...}}}}$

The infinite continued fraction can be written, $\sqrt{2} = [1; (2)]$ indicates that 2 repeats <i>ad infinitum</i>. In a similar way, $\sqrt{23} = [4; (1, 3, 1, 8)]$.
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 $\sqrt{2}$.

$1 + \dfrac{1}{2} = \dfrac{3}{2}\\\\
1 + \dfrac{1}{2 + \dfrac{1}{2}} = \dfrac{7}{5}\\\\
1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2}}} = \dfrac{17}{12}\\\\
1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2}}}} = \dfrac{41}{29}$

Hence the sequence of the first ten convergents for $\sqrt{2}$ are:

$1, \dfrac{3}{2}, \dfrac{7}{5}, \dfrac{17}{12}, \dfrac{41}{29}, \dfrac{99}{70}, \dfrac{239}{169}, \dfrac{577}{408}, \dfrac{1393}{985}, \dfrac{3363}{2378}, ...$

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 <var>e</var> are:

$2, 3, \dfrac{8}{3}, \dfrac{11}{4}, \dfrac{19}{7}, \dfrac{87}{32}, \dfrac{106}{39}, \dfrac{193}{71}, \dfrac{1264}{465}, \dfrac{1457}{536}, ...$

The sum of digits in the numerator of the 10<sup>th</sup> convergent is $1 + 4 + 5 + 7 = 17$.

Find the sum of digits in the numerator of the 100<sup>th</sup> convergent of the continued fraction for <var>e</var>.

</section>

## Instructions
<section id='instructions'>

</section>

## Tests
<section id='tests'>

```yml
tests:
  - text: <code>convergentsOfE()</code> should return a number.
    testString: assert(typeof convergentsOfE() === 'number');
  - text: <code>convergentsOfE()</code> should return 272.
    testString: assert.strictEqual(convergentsOfE(), 272);

```

</section>

## Challenge Seed
<section id='challengeSeed'>

<div id='js-seed'>

```js
function convergentsOfE() {

  return true;
}

convergentsOfE();
```

</div>



</section>

## Solution
<section id='solution'>

```js
// solution required
```

</section>