## 2018年10月21日日曜日

### 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)

1. $\begin{array}{}\underset{n\to \infty }{\mathrm{lim}}{\left(\frac{\left(2n\right)!}{{n}^{2n}}\right)}^{\frac{1}{n}}\\ =\underset{n\to \infty }{\mathrm{lim}}{\left(\frac{{2}^{2n}\left(2n\right)!}{{2}^{2n}{n}^{2n}}\right)}^{\frac{1}{2n}·}\\ =\underset{n\to \infty }{\mathrm{lim}}{\left(\frac{\left(2n\right)!}{{\left(2n\right)}^{2n}}·{2}^{2n}\right)}^{\frac{1}{2n}·2}\\ =\underset{n\to \infty }{\mathrm{lim}}{\left({\left(\frac{\left(2n\right)!}{{\left(2n\right)}^{2n}}\right)}^{\frac{1}{2n}}\right)}^{2}{2}^{2}\\ =\frac{4}{{e}^{2}}\end{array}$

2. $\begin{array}{}\underset{n\to \infty }{\mathrm{lim}}{\left(\frac{\left(2n\right)!\left(5n\right)!}{{n}^{4n}\left(3n\right)!}\right)}^{\frac{1}{n}}\\ =\underset{n\to \infty }{\mathrm{lim}}{\left(\frac{\left(2n\right)!\left(5n\right)!}{{n}^{2n}{n}^{2n}\left(3n\right)!}\right)}^{\frac{1}{n}}\\ =\underset{n\to \infty }{\mathrm{lim}}{\left(\frac{{2}^{2n}}{{2}^{2n}}·\frac{\left(2n\right)!}{{n}^{2n}}·\frac{{2}^{2n}}{{2}^{2n}}·\frac{\left(2n\right)!}{\left(2n\right)!}·\frac{1}{{n}^{2n}}·\frac{{\left(3n\right)}^{3n}}{{\left(3n\right)}^{3n}}·\frac{1}{\left(3n\right)!}·\frac{{\left(5n\right)}^{5n}}{{\left(5n\right)}^{5n}}·\left(5n\right)!\right)}^{\frac{1}{n}}\\ =\underset{n\to \infty }{\mathrm{lim}}{\left({2}^{2n}·\frac{\left(2n\right)!}{{\left(2n\right)}^{2n}}·\frac{{2}^{2n}}{\left(2n\right)!}·\frac{\left(2n\right)!}{{\left(2n\right)}^{2n}}·\frac{1}{{\left(3n\right)}^{3n}}·\frac{{\left(3n\right)}^{3n}}{\left(3n\right)!}·{\left(5n\right)}^{5n}·\frac{\left(5n\right)!}{{\left(5n\right)}^{5n}}\right)}^{\frac{1}{n}}\\ ={2}^{4}\underset{n\to \infty }{\mathrm{lim}}{\left(\frac{\left(2n\right)!}{{\left(2n\right)}^{2n}}·\frac{\left(2n\right)!}{{\left(2n\right)}^{2n}}·\frac{{\left(3n\right)}^{3n}}{\left(3n\right)!}·\frac{\left(5n\right)!}{{\left(5n\right)}^{5n}}·\frac{{\left(2n\right)}^{2n}}{\left(2n\right)!}·\frac{{\left(5n\right)}^{5n}}{{\left(2n\right)}^{2n}{\left(3n\right)}^{3n}}\right)}^{\frac{1}{n}}\\ ={2}^{4}\frac{1}{{e}^{2}}·\frac{1}{{e}^{2}}·{e}^{3}·\frac{1}{{e}^{5}}·{e}^{2}·\frac{{5}^{5}}{{2}^{2}·{3}^{3}}\\ =\frac{{2}^{2}{5}^{5}}{{3}^{3}{e}^{4}}\end{array}$

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()


$./sample2.py 2. (a) -2 4⋅ℯ (b) -4 12500⋅ℯ ───────── 27$


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,
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])
let yscale = d3.scaleLinear()
.domain([y1, y2])

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();