数学 - Python - JavaScript - 解析学 - 積分 - いくつかの計算練習 - (n!)^(1/n)の計算(指数関数、逆数、階乗、累乗、極限)

解析入門 原書第3版
原著
楽天ブックス(Kobo)

学習環境

• Surface Go、タイプ カバー、ペン(端末)
• Windows 10 Pro (OS)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ(iOS))
• 参考書籍
  • Pythonからはじめる数学入門 (楽天ブックス(Kobo))
  • JavaScriptリファレンス 第6版 (楽天ブックス(Kobo))
  • インタラクティブ・データビジュアライゼーション (楽天ブックス(Kobo))

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第3部(積分)、第12章(いくつかの計算練習)、1((n!)^(1/n)の計算)の問題2.を取り組んでみる。

2. 
   
   1. 
      
      $\begin{array}{}\underset{n\to \; <!--\; rightwards\; arrow\; -->\mathrm{\infty \; <!--\; infinity\; -->}}{\lim }{\left(\frac{\left(2n\right)!}{n^{2n}}\right)}^{\frac{1}{n}}\\ =\underset{n\to \; <!--\; rightwards\; arrow\; -->\mathrm{\infty \; <!--\; infinity\; -->}}{\lim }{\left(\frac{2^{2n}\left(2n\right)!}{2^{2n}{n}^{2n}}\right)}^{\frac{1}{2n}·\; <!--\; middle\; dot\; -->}\\ =\underset{n\to \; <!--\; rightwards\; arrow\; -->\mathrm{\infty \; <!--\; infinity\; -->}}{\lim }{\left(\frac{\left(2n\right)!}{{\left(2n\right)}^{2n}}·\; <!--\; middle\; dot\; -->2^{2n}\right)}^{\frac{1}{2n}·\; <!--\; middle\; dot\; -->2}\\ =\underset{n\to \; <!--\; rightwards\; arrow\; -->\mathrm{\infty \; <!--\; infinity\; -->}}{\lim }{\left({\left(\frac{\left(2n\right)!}{{\left(2n\right)}^{2n}}\right)}^{\frac{1}{2n}}\right)}^2{2}^2\\ =\frac{4}{e^2}\end{array}$
   2. 
      
      $\begin{array}{}\underset{n\to \; <!--\; rightwards\; arrow\; -->\mathrm{\infty \; <!--\; infinity\; -->}}{\lim }{\left(\frac{\left(2n\right)!\left(5n\right)!}{n^{4n}\left(3n\right)!}\right)}^{\frac{1}{n}}\\ =\underset{n\to \; <!--\; rightwards\; arrow\; -->\mathrm{\infty \; <!--\; infinity\; -->}}{\lim }{\left(\frac{\left(2n\right)!\left(5n\right)!}{n^{2n}{n}^{2n}\left(3n\right)!}\right)}^{\frac{1}{n}}\\ =\underset{n\to \; <!--\; rightwards\; arrow\; -->\mathrm{\infty \; <!--\; infinity\; -->}}{\lim }{\left(\frac{2^{2n}}{2^{2n}}·\; <!--\; middle\; dot\; -->\frac{\left(2n\right)!}{n^{2n}}·\; <!--\; middle\; dot\; -->\frac{2^{2n}}{2^{2n}}·\; <!--\; middle\; dot\; -->\frac{\left(2n\right)!}{\left(2n\right)!}·\; <!--\; middle\; dot\; -->\frac{1}{n^{2n}}·\; <!--\; middle\; dot\; -->\frac{{\left(3n\right)}^{3n}}{{\left(3n\right)}^{3n}}·\; <!--\; middle\; dot\; -->\frac{1}{\left(3n\right)!}·\; <!--\; middle\; dot\; -->\frac{{\left(5n\right)}^{5n}}{{\left(5n\right)}^{5n}}·\; <!--\; middle\; dot\; -->\left(5n\right)!\right)}^{\frac{1}{n}}\\ =\underset{n\to \; <!--\; rightwards\; arrow\; -->\mathrm{\infty \; <!--\; infinity\; -->}}{\lim }{\left(2^{2n}·\; <!--\; middle\; dot\; -->\frac{\left(2n\right)!}{{\left(2n\right)}^{2n}}·\; <!--\; middle\; dot\; -->\frac{2^{2n}}{\left(2n\right)!}·\; <!--\; middle\; dot\; -->\frac{\left(2n\right)!}{{\left(2n\right)}^{2n}}·\; <!--\; middle\; dot\; -->\frac{1}{{\left(3n\right)}^{3n}}·\; <!--\; middle\; dot\; -->\frac{{\left(3n\right)}^{3n}}{\left(3n\right)!}·\; <!--\; middle\; dot\; -->{\left(5n\right)}^{5n}·\; <!--\; middle\; dot\; -->\frac{\left(5n\right)!}{{\left(5n\right)}^{5n}}\right)}^{\frac{1}{n}}\\ =2^4\underset{n\to \; <!--\; rightwards\; arrow\; -->\mathrm{\infty \; <!--\; infinity\; -->}}{\lim }{\left(\frac{\left(2n\right)!}{{\left(2n\right)}^{2n}}·\; <!--\; middle\; dot\; -->\frac{\left(2n\right)!}{{\left(2n\right)}^{2n}}·\; <!--\; middle\; dot\; -->\frac{{\left(3n\right)}^{3n}}{\left(3n\right)!}·\; <!--\; middle\; dot\; -->\frac{\left(5n\right)!}{{\left(5n\right)}^{5n}}·\; <!--\; middle\; dot\; -->\frac{{\left(2n\right)}^{2n}}{\left(2n\right)!}·\; <!--\; middle\; dot\; -->\frac{{\left(5n\right)}^{5n}}{{\left(2n\right)}^{2n}{\left(3n\right)}^{3n}}\right)}^{\frac{1}{n}}\\ =2^4\frac{1}{e^2}·\; <!--\; middle\; dot\; -->\frac{1}{e^2}·\; <!--\; middle\; dot\; -->e^3·\; <!--\; middle\; dot\; -->\frac{1}{e^5}·\; <!--\; middle\; dot\; -->e^2·\; <!--\; middle\; dot\; -->\frac{5^5}{2^2·\; <!--\; middle\; dot\; -->3^3}\\ =\frac{2^2{5}^5}{3^3{e}^4}\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, exp, factorial, oo, plot, floor

print('2.')

n = symbols('n', integer=True)
ts = [(factorial(2 * n), n ** (2 * n)),
      (factorial(2 * n) * factorial(5 * n), (n ** (4 * n) * factorial(3 * n)))]

for i, (num, den) in enumerate(ts):
    print(f'({chr(ord("a") + i)})')
    pprint(((num / den) ** (1 / n)).limit(n, oo).doit())
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample2.py
2.
(a)
   -2
4⋅ℯ  

(b)
       -4
12500⋅ℯ  
─────────
    27   

$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.01">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="20">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="20">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample2.js"></script>

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    input_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let factorial = n => n < 1 ? 1 : n * factorial(n - 1),
    fns = [
        [x => {
            let n = Math.floor(x);
            
            return (factorial(2 * n) / n ** (2 * n)) ** (1 / n);
        }, 'red'],
        [x => {
            let n = Math.floor(x);

            return  ((factorial(2 * n) * factorial(5 * n)) /
                     (n ** (4 * n) * factorial(3 * n))) ** (1 / n);
        }, 'green']
    ];

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    
    
    let points = [],
        lines = [[x1, 4 / Math.exp(2), x2, 4 / Math.exp(2), 'blue'],
                 [x1, 4 * 5 ** 5 / (27 * Math.exp(4)),
                  x2, 4 * 5 ** 5 / (27 * Math.exp(4)), 'orange']];
    
    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, color]);
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');

    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    [fns].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =