数学 - 解析学 - JavaScript - 数列と級数 - 数列の収束条件(部分列極限、単調増加、単調減少、有界)

解析入門〈1〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax
• Emacs (Text Editor)
• JavaScript (プログラミング言語)
• D3.js (JavaScript Library)
• 参考書籍
  • JavaScript 第6版 (David Flanagan(著)、村上 列(翻訳)、オライリージャパン)
  • JavaScriptリファレンス 第6版(David Flanagan(著)、木下 哲也(翻訳)、オライリージャパン)
  • インタラクティブ・データビジュアライゼーション(Scott Murray (著)、長尾 高弘 (翻訳)、オライリージャパン)

解析入門〈1〉(松坂 和夫(著)、岩波書店)の第2章(数列と級数)、2.2(数列の収束条件)、問題2.2、4-a、b.を取り組んでみる。

問題2.2、4-a、b.

1. 
   $\begin{array}{l}{a}_n-a_{n+2}\\ =a_n-\frac{a_{n+1}+\alpha }{a_{n+1}+1}\\ =a_n-\frac{\frac{a_n+\alpha }{a_n+1}+\alpha }{\frac{a_n+\alpha }{a_n+1}+1}\\ =a_n-\frac{a_n+\alpha +\alpha \left(a_n+1\right)}{a_n+1}·\frac{a_n+1}{a_n+\alpha +a_n+1}\\ =a_n-\frac{\left(1+\alpha \right)a_n+2\alpha }{2a_n+\alpha +1}\\ =\frac{2a_n{}^2+\left(\alpha +1\right)a_n-\left(\alpha +1\right)a_n-2\alpha }{2a_n+\alpha +1}\\ =\frac{2\left(a_n{}^2-\alpha \right)}{2a_n+\alpha +1}\\ \\ a_{n+1}-\sqrt{\alpha }\\ =\frac{a_n+\alpha }{a_n+1}-\sqrt{\alpha }\\ =\frac{a_n+\alpha -\sqrt{\alpha }{a}_n-\sqrt{\alpha }}{a_n+1}\\ =\frac{\left(1-\sqrt{\alpha }\right)a_n+\sqrt{\alpha }\left(\sqrt{\alpha }-1\right)}{a_n+1}\\ =\frac{\left(\sqrt{\alpha }-1\right)\left(\sqrt{\alpha }-a_n\right)}{a_n+1}\\ \\ a_n<\sqrt{\alpha }\Rightarrow a_n<a_{n+2}\\ a_n>\sqrt{a}\Rightarrow a_n>a_{n+2}\\ \\ a_1<\sqrt{\alpha }\\ a_2-\sqrt{\alpha }=\frac{\left(\sqrt{\alpha }-1\right)\left(\sqrt{\alpha }-a_1\right)}{a_1+1}>0\\ a_2>\sqrt{\alpha }\end{array}$
2. 
   $\begin{array}{l}{a}_{2n-1}<\sqrt{\alpha }\\ a_{2n}>\sqrt{\alpha }\\ \\ a_{n+2}=\frac{\frac{a_n+\alpha }{a_n+1}+\alpha }{\frac{a_n+\alpha }{a_n+1}+1}\\ \beta =\frac{\frac{\beta +\alpha }{\beta +1}+\alpha }{\frac{\beta +\alpha }{\beta +1}+1}\\ \beta =\frac{\left(1+\alpha \right)\beta +2\alpha }{2\beta +\alpha +1}\\ 2{\beta }^2+\alpha \beta +\beta =\beta +\alpha \beta +2\alpha \\ {\beta }^2=\alpha \\ \beta =\sqrt{\alpha }\end{array}$

HTML5

<div id="graph0"></div>
<div id="output0"></div>
<label for="alpha0">α = </label>
<input id="alpha0" type="number" min="1" value="5">
<label for="a1">a1 = </label>
<input id="a1" type="number" min="0" value="2">
<label for="n0">n = </label>
<input id="n0" type="number" min="1" step="1" value="10">


<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="sample4.js"></script>

JavaScript

{
    let div_graph = document.querySelector('#graph0'),
        div_output = document.querySelector('#output0'),
        input_alpha = document.querySelector('#alpha0'),
        input_a = document.querySelector('#a1'),
        input_n = document.querySelector('#n0'),
        inputs = [input_alpha, input_a, input_n],
        width = 600,
        height = 600,
        padding = 50;

    let f = (alpha, prev) => {
        return (prev + alpha) / (prev + 1);
    };
    let getPoints = (alpha, a1, n) => {
        let prev = a1,
            result = [[1, prev]];

        for (let i = 2; i <= n; i += 1) {
            let next = f(alpha, prev);
            result.push([i, next]);
            prev = next;
        }
        return result;
    }
    let plot = () => {
        let alpha = parseFloat(input_alpha.value),
            a1 = parseFloat(input_a.value);

        if (alpha > 1 && 0 < a1 && a1 < Math.sqrt(alpha)) {
            let n = parseInt(input_n.value, 10),
                points = getPoints(alpha, a1, n);

            console.log(points);
            let xscale = d3.scaleLinear()
                .domain([1, n])
                .range([padding, width - padding])
            let yscale = d3.scaleLinear()
                .domain([points[0][1] * 0.9, points[1][1] * 1.1])
                .range([height - padding, padding]);

            console.log(points.map((x) => x[1]));
            div_graph.innerHTML = '';
            let svg = d3.select('#graph0')
                .append('svg')
                .attr('width', width)
                .attr('height', height);

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

            let xaxis = d3.axisBottom().scale(xscale);
            let yaxis = d3.axisLeft().scale(yscale);

            svg.append('line')
                .attr('x1', xscale(1))
                .attr('y1', yscale(Math.sqrt(alpha)))
                .attr('x2', xscale(n))
                .attr('y2', yscale(Math.sqrt(alpha)))
                .attr('stroke', 'red');
            
            svg.append('g')
                .attr('transform', `translate(0, ${height - padding})`)
                .call(xaxis);
            svg.append('g')
                .attr('transform', `translate(${padding}, 0)`)
                .call(yaxis);

            div_output.innerHTML =
                points.map((x) => `${x[0]}: ${x[1]}`).join('<br>') +
                `<br>√α(Math.sqrt(${alpha})): ${Math.sqrt(alpha)}`;
            
        } else {
            div_output.innerHTML = '注意: α > 1, 0 < a1 < √α';
        }
    };

    inputs.forEach((input) => input.onchange = plot);
    plot();

}

1: 2
2: 2.3333333333333335
3: 2.2
4: 2.25
5: 2.230769230769231
6: 2.238095238095238
7: 2.235294117647059
8: 2.2363636363636363
9: 2.235955056179775
10: 2.236111111111111
√α(Math.sqrt(5)): 2.23606797749979

α = a1 = n =