数学 - Python - JavaScript - 解析学 - 積分 - いくつかの計算練習 - (n!)^(1/n)の計算(階乗、累乗、指数関数、不等号、対数関数の定積分、上方和および下方和との比較)

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
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

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)、定理1の証明を取り組んでみる。

0. 
   
   $\begin{array}{}{\int \; <!--\; integral\; -->}_1^n\log x\mathrm{dx}\\ ={\left[x\log x\right]}_1^n-\underset{1}{\overset{n}{\int \; <!--\; integral\; -->}}1\mathrm{dx}\\ =n\log n-{\left[x\right]}_1^n\\ =n\log n-n+1\end{array}$

   上方和。

   $\underset{k=2}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}\log k$

   下方和。

   $\underset{k=1}{\overset{n-1}{\sum \; <!--\; n-ary\; summation\; -->}}\log k$

   比較。

   $\underset{k=1}{\overset{n-1}{\sum \; <!--\; n-ary\; summation\; -->}}\log k\le \; <!--\; less-than\; or\; equal\; to\; -->n\log n-n+1\le \; <!--\; less-than\; or\; equal\; to\; -->\underset{k=2}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}\log k$

   e の肩にのせる。（指数関数）

   $\begin{array}{}{e}^{\underset{k=1}{\overset{n-1}{\sum \; <!--\; n-ary\; summation\; -->}}\log k}\le \; <!--\; less-than\; or\; equal\; to\; -->e^{\left(n\log n-n+1\right)}\le \; <!--\; less-than\; or\; equal\; to\; -->e^{\underset{k=2}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}\log k}\\ \underset{k=1}{\overset{n-1}{\prod \; <!--\; n-ary\; product\; -->}}{e}^{\log k}\le \; <!--\; less-than\; or\; equal\; to\; -->n^n{e}^{-n}e\le \; <!--\; less-than\; or\; equal\; to\; -->\underset{k=2}{\overset{n}{\prod \; <!--\; n-ary\; product\; -->}}{e}^{\log k}\\ \left(n-1\right)!\le \; <!--\; less-than\; or\; equal\; to\; -->n^n{e}^{-n}e\le \; <!--\; less-than\; or\; equal\; to\; -->n!\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, log, factorial, exp, E, plot

print('定理1の証明.')

x = symbols('x', real=True)
n = symbols('n', integer=True, nonnegative=True)
I = Integral(log(x), (x, 1, n))

for t in [I, I.doit()]:
    pprint(t)
    print()

l = factorial(n - 1)
c = n ** n * exp(-n) * E
r = factorial(n)

for m in range(1, 10):
    print(f'n = {m}')
    d = {n:m}
    lm = l.subs(d)
    cm = c.subs(d)
    rm = r.subs(d)
    for t in [lm, cm, rm, lm <= cm <= rm]:
        pprint(t)
        print()

p = plot(l, c, r, (n, 1, 5), legend=True, show=False)
colors = ['red', 'green', 'blue']
for i, color in enumerate(colors):
    p[i].line_color = color
p.save('sample0.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample0.py
定理1の証明.
n          
⌠          
⎮ log(x) dx
⌡          
1          

n⋅log(n) - n + 1

n = 1
1

1

1

True

n = 2
1

   -1
4⋅ℯ  

2

True

n = 3
2

    -2
27⋅ℯ  

6

True

n = 4
6

     -3
256⋅ℯ  

24

True

n = 5
24

      -4
3125⋅ℯ  

120

True

n = 6
120

       -5
46656⋅ℯ  

720

True

n = 7
720

        -6
823543⋅ℯ  

5040

True

n = 8
5040

          -7
16777216⋅ℯ  

40320

True

n = 9
40320

           -8
387420489⋅ℯ  

362880

True

$

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="1">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="50">

<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="sample0.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 => factorial(Math.floor(x) - 1), 'red'],
           [x => {
               let n = Math.floor(x);

               return (n ** n) * Math.exp(-n) * Math.E;
           }, 'green'],
           [x => factorial(Math.floor(x)), 'blue']];

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 = [[1, y1, 1, y2, '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 =