chore(i8n,learn): processed translations

curriculum/challenges/espanol/10-coding-interview-prep/project-euler/problem-65-convergents-of-e.md

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+---
+id: 5900f3ad1000cf542c50fec0
+title: 'Problem 65: Convergents of e'
+challengeType: 5
+forumTopicId: 302177
+dashedName: problem-65-convergents-of-e
+---
+
+# --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 *ad infinitum*. 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 `e` 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 `e`.
+
+# --hints--
+
+`convergentsOfE()` should return a number.
+
+```js
+assert(typeof convergentsOfE() === 'number');
+```
+
+`convergentsOfE()` should return 272.
+
+```js
+assert.strictEqual(convergentsOfE(), 272);
+```
+
+# --seed--
+
+## --seed-contents--
+
+```js
+function convergentsOfE() {
+
+  return true;
+}
+
+convergentsOfE();
+```
+
+# --solutions--
+
+```js
+// solution required
+```