gaspersic/freeCodeCamp
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

chore(i18n,learn): processed translations (#45192)

Browse Source
This commit is contained in:
camperbot
2022-02-19 20:11:19 +05:30
committed by GitHub
parent 75f3278c06
commit 2d3ac85ebf
49 changed files with 444 additions and 492 deletions
Download Patch File Download Diff File Expand all files Collapse all files

26
curriculum/challenges/italian/10-coding-interview-prep/project-euler/problem-65-convergents-of-e.md
View File

@@ -1,6 +1,6 @@
---
id: 5900f3ad1000cf542c50fec0
title: 'Problem 65: Convergents of e'
title: 'Problema 65: convergenti di e'
challengeType: 5
forumTopicId: 302177
dashedName: problem-65-convergents-of-e
@@ -8,59 +8,59 @@ dashedName: problem-65-convergents-of-e
# --description--
The square root of 2 can be written as an infinite continued fraction.
La radice quadrata di 2 può essere scritta come una frazione continuata infinita.
$\\sqrt{2} = 1 + \\dfrac{1}{2 + \\dfrac{1}{2 + \\dfrac{1}{2 + \\dfrac{1}{2 + ...}}}}$
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 $\\sqrt{2}$.
La frazione continuata infinita può essere scritta, $\\sqrt{2} = \[1; (2)]$ indica che 2 ripete *ad infinitum*. In modo simile, $\\sqrt{23} = \[4; (1, 3, 1, 8)]$. La sequenza di valori parziali di frazioni continuati delle radici quadrate provvede l'approssimazione razionale migliore. Consideriamo le convergenze con $\\sqrt{2}$.
$1 + \\dfrac{1}{2} = \\dfrac{3}{2}\\\\ 1 + \\dfrac{1}{2 + \\dfrac{1}{2}} = \\dfrac{7}{5}\\\\ 1 + \\dfrac{1}{2 + \\dfrac{1}{2 + \\dfrac{1}{2}}} = \\dfrac{17}{12}\\\\ 1 + \\dfrac{1}{2 + \\dfrac{1}{2 + \\dfrac{1}{2 + \\dfrac{1}{2}}}} = \\dfrac{41}{29}$
Hence the sequence of the first ten convergents for $\\sqrt{2}$ are:
Quindi la sequenza delle prime dieci convergenze per $\\sqrt{2}$ sono:
$1, \\dfrac{3}{2}, \\dfrac{7}{5}, \\dfrac{17}{12}, \\dfrac{41}{29}, \\dfrac{99}{70}, \\dfrac{239}{169}, \\dfrac{577}{408}, \\dfrac{1393}{985}, \\dfrac{3363}{2378}, ...$
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:
La cosa sorprendente è che la più importante costante matematica, $e = \[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, ... , 1, 2k, 1, ...]$. I primi dieci termini della sequenza convergente di `e` sono:
$2, 3, \\dfrac{8}{3}, \\dfrac{11}{4}, \\dfrac{19}{7}, \\dfrac{87}{32}, \\dfrac{106}{39}, \\dfrac{193}{71}, \\dfrac{1264}{465}, \\dfrac{1457}{536}, ...$
The sum of digits in the numerator of the 10<sup>th</sup> convergent is $1 + 4 + 5 + 7 = 17$.
La somma delle digite del numeratore del 10<sup>o</sup> convergente è $1 + 4 + 5 + 7 = 17$.
Find the sum of digits in the numerator of the `n`<sup>th</sup> convergent of the continued fraction for `e`.
Trova la somma delle cifre nel numeratore per il `n`<sup>o</sup> convergente della frazione continuata di `e`.
# --hints--
`convergentsOfE(10)` should return a number.
`convergentsOfE(10)` dovrebbe restituire un numero.
```js
assert(typeof convergentsOfE(10) === 'number');
```
`convergentsOfE(10)` should return `17`.
`convergentsOfE(10)` dovrebbe restituire `17`.
```js
assert.strictEqual(convergentsOfE(10), 17);
```
`convergentsOfE(30)` should return `53`.
`convergentsOfE(30)` dovrebbe restituire `53`.
```js
assert.strictEqual(convergentsOfE(30), 53);
```
`convergentsOfE(50)` should return `91`.
`convergentsOfE(50)` dovrebbe restituire `91`.
```js
assert.strictEqual(convergentsOfE(50), 91);
```
`convergentsOfE(70)` should return `169`.
`convergentsOfE(70)` dovrebbe restituire `169`.
```js
assert.strictEqual(convergentsOfE(70), 169);
```
`convergentsOfE(100)` should return `272`.
`convergentsOfE(100)` dovrebbe restituire `272`.
```js
assert.strictEqual(convergentsOfE(100), 272);