{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 01 収束\n", "\n", "黒木玄\n", "\n", "2018-04-18, 2023-06-22\n", "\n", "* Copyright 2018, 2023 Gen Kuroki\n", "* License: MIT https://opensource.org/licenses/MIT\n", "* Repository: https://github.com/genkuroki/Calculus\n", "\n", "このファイルは次の場所できれいに閲覧できる:\n", "\n", "* http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/01%20convergence.ipynb\n", "\n", "* https://genkuroki.github.io/documents/Calculus/01%20convergence.pdf\n", "\n", "このファイルは Julia Box で利用できる.\n", "\n", "自分のパソコンにJulia言語をインストールしたい場合には\n", "\n", "* [WindowsへのJulia言語のインストール](http://nbviewer.jupyter.org/gist/genkuroki/81de23edcae631a995e19a2ecf946a4f)\n", "\n", "* [Julia v1.1.0 の Windows 8.1 へのインストール](https://nbviewer.jupyter.org/github/genkuroki/msfd28/blob/master/install.ipynb)\n", "\n", "を参照せよ. 前者は古く, 後者の方が新しい.\n", "\n", "論理的に完璧な説明をするつもりはない. 細部のいい加減な部分は自分で訂正・修正せよ.\n", "\n", "$\n", "\\newcommand\\eps{\\varepsilon}\n", "\\newcommand\\ds{\\displaystyle}\n", "\\newcommand\\Z{{\\mathbb Z}}\n", "\\newcommand\\Q{{\\mathbb Q}}\n", "\\newcommand\\R{{\\mathbb R}}\n", "\\newcommand\\C{{\\mathbb C}}\n", "\\newcommand\\QED{\\text{□}}\n", "\\newcommand\\root{\\sqrt}\n", "$" ] }, { "cell_type": "markdown", "metadata": { "toc": true }, "source": [ "

目次

\n", "
" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "using Base.MathConstants\n", "using Base64\n", "using Printf\n", "using Statistics\n", "const e = ℯ\n", "endof(a) = lastindex(a)\n", "linspace(start, stop, length) = range(start, stop, length=length)\n", "\n", "using Plots\n", "#gr(); ENV[\"PLOTS_TEST\"] = \"true\"\n", "#clibrary(:colorcet)\n", "default(fmt=:png)\n", "#clibrary(:misc)\n", "\n", "function pngplot(P...; kwargs...)\n", " sleep(0.1)\n", " pngfile = tempname() * \".png\"\n", " savefig(plot(P...; kwargs...), pngfile)\n", " showimg(\"image/png\", pngfile)\n", "end\n", "pngplot(; kwargs...) = pngplot(plot!(; kwargs...))\n", "\n", "showimg(mime, fn) = open(fn) do f\n", " base64 = base64encode(f)\n", " display(\"text/html\", \"\"\"\"\"\")\n", "end\n", "\n", "using SymPy\n", "#sympy.init_printing(order=\"lex\") # default\n", "#sympy.init_printing(order=\"rev-lex\")\n", "\n", "using SpecialFunctions\n", "using QuadGK" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# Override the Base.show definition of SymPy.jl:\n", "# https://github.com/JuliaPy/SymPy.jl/blob/29c5bfd1d10ac53014fa7fef468bc8deccadc2fc/src/types.jl#L87-L105\n", "\n", "@eval SymPy function Base.show(io::IO, ::MIME\"text/latex\", x::SymbolicObject)\n", " print(io, as_markdown(\"\\\\displaystyle \" * sympy.latex(x, mode=\"plain\", fold_short_frac=false)))\n", "end\n", "@eval SymPy function Base.show(io::IO, ::MIME\"text/latex\", x::AbstractArray{Sym})\n", " function toeqnarray(x::Vector{Sym})\n", " a = join([\"\\\\displaystyle \" * sympy.latex(x[i]) for i in 1:length(x)], \"\\\\\\\\\")\n", " \"\"\"\\\\left[ \\\\begin{array}{r}$a\\\\end{array} \\\\right]\"\"\"\n", " end\n", " function toeqnarray(x::AbstractArray{Sym,2})\n", " sz = size(x)\n", " a = join([join(\"\\\\displaystyle \" .* map(sympy.latex, x[i,:]), \"&\") for i in 1:sz[1]], \"\\\\\\\\\")\n", " \"\\\\left[ \\\\begin{array}{\" * repeat(\"r\",sz[2]) * \"}\" * a * \"\\\\end{array}\\\\right]\"\n", " end\n", " print(io, as_markdown(toeqnarray(x)))\n", "end" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 収束の定義" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 数列の収束の定義\n", "\n", "**定義:** 数列 $a_n$ が $n\\to\\infty$ で $\\alpha$ に**収束する**とは, 任意の $\\eps>0$ に対して, ある番号 $N$ が存在して, $N$ 以降のすべての番号 $n$ について $|a_n - \\alpha| < \\eps$ が成立することである. $\\QED$\n", "\n", "$a_n$ が $\\alpha$ に収束するとき, $a_n$ の極限の値を $\\displaystyle \\lim_{n\\to\\infty} a_n = \\alpha$ と定義する. 収束しないときには極限の値は定義されない.\n", "\n", "$a_n$ が $\\alpha$ に収束することを $a_n\\to \\alpha$ と書くことにする.\n", "\n", "上の定義に基けば, 数列 $a_n$ が $n\\to\\infty$ で $\\alpha$ に**収束しない**ことは, ある $\\eps>0$ が存在して, 任意の番号 $N$ について, ある $n\\geqq N$ で $|a_n-\\alpha|\\geqq\\eps$ を満たすものが存在することだと言い換えられる. \n", "\n", "どのような数にも収束しない数列は**発散**しているという. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "以上は所謂 $\\eps$-$N$ 論法による数列の収束の定義である.\n", "\n", "**勉強の仕方について1:** このノートから始まる一連のノート群では $\\eps$-$N$ 論法や $\\eps$-$\\delta$ 論法による詳しい証明は**多くの場合に扱わない予定**である. \n", "\n", "しかし, 以下では数列の収束について $\\eps$-$N$ について詳しく説明しておくことにする. $\\eps$-$N$ および $\\eps$-$\\delta$ の方が易しく扱える場合は結構多いので教養として身に付けておいて損はないと思われる. \n", "\n", "特に解析学の知識を信頼できる数値計算の遂行に役立てたいと思っている人は, どこかの段階で ε-N と ε-δ の考え方をマスターしておくべきである. なぜならば, **ε-N と ε-δ は「誤差の評価をまじめに行うこと」そのもの**だからである. 単に数学を応用するだけならば, ε-N と ε-δ は必要ないという考え方は誤りである.\n", "\n", "初心者のために, 下の方には ε-N を使わない解答例1と ε-N を使った解答例2の両方を解説した場合を複数書いておいた. それらの解答を比較することは, 収束に関する直観と論理の整備に役に立つだろう.\n", "\n", "**しかし, 無理はいけない.** \n", "\n", "初心者のうちは ε-N および ε-δ を使った証明を無理してフォローする必要はない.\n", "\n", "**すぐに理解できそうもないならば, 「適当に証明を飛ばしながら先に進み, 必要に応じて何度でも前に戻ることを繰り返しながら膨大な時間をかければいつかは理解できる」のように考えるのがよいと思う.** $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**勉強の仕方について2:** 筆者の個人的な経験では, 初心者のうちは, $\\eps$-$N$ や $\\eps$-$\\delta$ の議論を書き下すときには, $\\forall$, $\\exists$ という記号を**使わない方がよい**と思う. より正確に言えば, $\\forall$, $\\exists$ という記号を使わずに説明できないと感じたら, $\\forall$, $\\exists$ という記号を使わない方がよい. 記号に頼る議論は非常によろしくない. 頼るべきなのは記号ではなく, 健全な直観である.\n", "\n", "特に解析学の議論を $\\forall$, $\\exists$ という記号を使った形式的な記号計算として理解しようとするのはやめた方がよい. 解析学のポイントは「近似の誤差にあたるものが何にどのように依存しているか」を見極めることにある場合が多い. \n", "\n", "「〇〇の大きさが△△に◇◇のように依存して決まっており, 大雑把には□□と評価される」というような考え方を滑らかにできるようになることを目標にするのがよい. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**数列の収束の正確な定義の1つの解釈:** 上の定義は数列 $a_n$ で $\\alpha$ の値を近似計算したい場合には次のような意味を持つ. \n", "\n", "(1) 計算可能な数列 $a_n$ は直接正確な値を計算できない数値 $\\alpha$ に収束することがわかっているとする.\n", "\n", "(2) $\\eps>0$ を任意に与え, 数列 $a_n$ の計算によって数値 $\\alpha$ を誤差 $\\eps$ 未満で計算したいとする.\n", "\n", "(3) もしも数列 $a_n$ が $\\alpha$ に収束しているならば, ある番号 $N$ が存在して, その $n\\geqq N$ とすれば $a_n$ と $\\alpha$ の差は $\\eps$ 未満になる.\n", "\n", "(4) そのとき, もしも $\\eps$ から $N$ を具体的に求めることができれば, $N$ 以上の $n$ に対する $a_n$ を実際に計算することによって, $\\alpha$ の値を誤差 $\\eps$ 未満で求めることができる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**例:** $\\alpha=\\sqrt{2}$ に対して, $a_n$ を $\\sqrt{2}$ を小数点以下第 $n$ 桁まで計算した結果だとする. 例えば, $a_1=1.4$, $a_2=1.41$, $a_3=1.414$, $\\ldots$. このとき, $|a_n-\\alpha|<10^{-n}$ なので, $\\alpha=\\sqrt{2}$ を誤差 $10^{-N}$ 未満で求めるためには $n\\geqq N$ に対する $a_n$ を求めればよい. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**数列が α に収束しないことの正確な定義の1つの解釈:** 数列 $a_n$ が $\\alpha$ に収束しないことは次のように言い換えられる: ある $\\eps>0$ が存在して, どんなに番号 $N$ を大きくしても, ある $n\\geqq N$ で $|a_n-\\alpha|\\geqq\\eps$ を満たすものが存在する.\n", "\n", "これは, 以下のような意味を持っている.\n", "\n", "(1) 数列 $a_n$ で数値 $\\alpha$ を近似計算したいとする.\n", "\n", "(2) しかし, $a_n$ は $\\alpha$ に収束していないとする.\n", "\n", "(3) その場合には, 許される誤差 $\\eps>0$ を小さく取り過ぎると, どんなに $N$ を大きくしても, $N$ 以上の $n$ の中に $a_n$ による $\\alpha$ の近似計算の誤差が許される誤差 $\\eps$ 以上になるものが存在する.\n", "\n", "もしも $a_n$ が $\\alpha$ に収束しているならば, どんなに許される誤差 $\\eps>0$ を小さくしても, 十分に $N$ を大きくすると, $N$ 以上の $n$ については常に $a_n$ による $\\alpha$ の近似計算の誤差が許される誤差 $\\eps$ 未満になる. すなわち, あるところから先の $a_n$ と $\\alpha$ の距離は $\\eps$ 未満になる.\n", "\n", "$a_n$ が $\\alpha$ に収束していなければ, 許される誤差 $\\eps>0$ を小さくし過ぎると, どんなに先の $a_N$ 以降の $a_N,a_{N+1},a_{N+2},\\ldots$ の中に誤差が $\\eps$ 以上になるものが含まれてしまう. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**例:** $a_n$ は上の例と同様に $\\alpha=\\sqrt{2}$ を小数点以下 $n$ 桁まで計算した結果であるとし, $b_n=a_n-(0.005+(-1)^n\\times 0.005)$ とおく. このとき, $b_{2k+1}=a_{2k+1}$ かつ $b_{2k}=a_n-0.01$ となる. だから, 奇数の $n$ についてのみ $b_n$ を考えれば $\\alpha=\\sqrt{2}$ に収束するが, 偶数の $n$ に対する $b_n$ は $\\sqrt{2}$ よりも $0.01$ 以上小さくなる. だから, どんなに番号 $N$ を大きくしても $b_N,b_{N+1},b_{N+2},\\ldots$ の中に $\\alpha=\\sqrt{2}$ との差が $0.01$ 以上になるものが存在する. これは $b_n$ が $\\alpha=\\sqrt{2}$ に収束しないということを意味している. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 数列 $a_n$ が $\\alpha$ に収束することと, 次の条件が同値であることを示せ:\n", "\n", "(☆) 任意の $\\eps>0$ に対して, $a_n$ と $\\alpha$ の距離が $\\eps$ 以上になる $n$ 達の個数は有限個しかない.\n", "\n", "数列 $a_n$ が $\\alpha$ に収束しないことと, 次の条件が同値であることを示せ:\n", "\n", "(★) ある $\\eps>0$ で, $a_n$ と $\\alpha$ からの距離が $\\eps$ 以上になる $n$ 達の個数が無限個になるものが存在する.\n", "\n", "**注意:** 同値なのでこれらの条件を収束する, しないの定義として採用してもよい.\n", "\n", "**解答例:** $a_n\\to\\alpha$ と仮定し, 任意に $\\eps>0$ を取る. 収束の定義より、ある番号 $N$ が存在して, $n\\geqq N$ ならば $|a_n-\\alpha|<\\eps$ となる. そのとき, $|a_n-\\alpha|\\geqq\\eps$ ならば $n0$ であるとする. $\\alpha$ から距離が $\\eps$ 以上の $a_n$ 達の個数は有限個なので, そのような $n$ の最大値が存在する. その最大値を $N-1$ と書くと, $n\\geqq N$ のとき $a_n$ と $\\alpha$ の距離は $\\eps$ 未満になる. これで $a_n\\to\\alpha$ が示された.\n", "\n", "条件(★)は条件(☆)の否定そのものであり, 条件(☆)は $a_n$ が $\\alpha$ に収束することの定義と同値だったので, 条件(★)は $a_n$ が $\\alpha$ に収束しないことと同値になる. \n", "\n", "これで示すべきことがすべて示されたが, 条件(★)と「ある $\\eps>0$ が存在して, 任意の番号 $N$ に対して, ある $n\\geqq N$ で $|a_n-\\alpha|\\geqq\\eps$ を満たすものが存在する」が同値であることも示しておこう.\n", "\n", "ある $\\eps>0$ が存在して, 任意の番号 $N$ に対して, ある $n\\geqq N$ で $|a_n-\\alpha|\\geqq\\eps$ を満たすものが存在すると仮定する. もしも $|a_n-\\alpha|\\geqq\\eps$ を満たす $n$ 達が有限個しかないとすると, そのような $n$ の最大値が存在し, そのとき $n$ がその最大値より大きければ常に $|a_n-\\alpha|<\\eps$ となってしまい仮定に反する。ゆえに $|a_n-\\alpha|\\geqq\\eps$ を満たす $n$ 達は無限個存在する. これで条件(★)が示された.\n", "\n", "条件(★)を仮定する. すなわち, ある $\\eps>0$ で, $a_n$ と $\\alpha$ からの距離が $\\eps$ 以上になる $n$ 達の個数が無限個になるものが存在すると仮定する. 任意に $N$ を取る. もしも $n\\geqq N$ ならば常に $|a_n-\\alpha|<\\eps$ となるならば, $|a_n-\\alpha|\\geqq\\eps$ となる $n$ 達の個数が $N-1$ 以下になってしまうので, 仮定に反する. ゆえに, ある $n\\geqq N$ で $|a_n-\\alpha|\\geqq\\eps$ となるものが存在する. これで, 「ある $\\eps>0$ が存在して, 任意の番号 $N$ に対して, ある $n\\geqq N$ で $|a_n-\\alpha|\\geqq\\eps$ を満たすものが存在する」という条件が示された. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題(三角不等式):** 実数 $A,B$ について\n", "\n", "$$\n", "|A\\pm B|\\leqq |A|+|B|, \\quad |A\\pm B|\\geqq |A|-|B|, \\quad |A\\pm B|\\geqq |B|-|A|\n", "$$\n", "\n", "となることを示せ. \n", "\n", "**解答例:**\n", "$$\n", "\\begin{aligned}\n", "&\n", "(|A|+|B|)^2 - |A\\pm B|^2 =\n", "|A|^2+2|A||B|+|B|^2 - (A^2\\pm 2AB+B^2) =\n", "2(|A||B| \\mp AB)\\geqq 0,\n", "\\end{aligned}\n", "$$\n", "\n", "より, $|A\\pm B|\\leqq |A|+|B|$ が得られる. その不等式で $A$ を $-(A\\pm B)$ で置き換えると, $|A|\\leqq |A\\pm B|+|B|$ が得られるので, $|A\\pm B|\\geqq |A|-|B|$ が得られる. その不等式で $A$ と $B$ の立場を取り換えると, $|A\\pm B|\\geqq |B|-|A|$ が得られる. $\\QED$\n", "\n", "**三角不等式のよくある使い方:** $|A-B|$ が小さいことを示すために, 三角不等式を\n", "\n", "$$\n", "|A-B| = |A-C+C-B|\\leqq |A-C|+|C-B|\n", "$$\n", "\n", "の形で用い, $|A-C|$ と $|C-B|$ が小さいことを示すという手段が非常によく使われる. ここで使われる $C$ は初等幾何の問題における補助線のようなものであり, うまい $C$ を見付けることが証明のポイントになることが多い. $\\QED$\n", "\n", "**三角不等式は空気のごとく多用される.**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 数列 $a_n$ が $\\alpha$ と $\\beta$ に収束しているならば, $\\alpha=\\beta$ となることを示せ. (これは同一の数列の収束先は唯一であることを意味している.)\n", "\n", "**解答例1:** $n\\to\\infty$ とすると,\n", "\n", "$$\n", "|\\beta-\\alpha|=|\\beta-a_n+a_n-\\alpha|\\leqq |\\beta-a_n|+|a_n-\\alpha|\\to 0.\n", "$$\n", "\n", "ゆえに $|\\beta-\\alpha|=0$ となるので, $\\beta=\\alpha$. $\\QED$\n", "\n", "**解答例2:** 任意に $\\eps>0$ を取る. $a_n$ は $\\alpha$ に収束しているので, ある番号 $N'$ が存在して, $n\\geqq N$ ならば $|a_n-\\alpha|<\\eps/2$ となる. $a_n$ は $\\beta$ にも収束しているので, ある番号 $N''$ が存在して, $n\\geqq N''$ ならば $|a_n-\\beta|<\\eps/2$ となる. ゆえに, $N=\\max\\{N',N''\\}$ とおくと, $n\\geqq N$ のとき,\n", "\n", "$$\n", "|\\beta-\\alpha| =\n", "|(a_n-\\alpha)-(a_n-\\beta)| \\leqq\n", "|a_n-\\alpha| + |a_n-\\beta|\n", "< \\frac{\\eps}{2}+\\frac{\\eps}{2} = \\eps.\n", "$$\n", "\n", "ここで $\\leqq$ では三角不等式を使った. $\\eps>0$ はいくらでも小さくできるので, $\\alpha=\\beta$ となる. $\\QED$\n", "\n", "**解説:** 以上の解答例は等式 $\\beta=\\alpha$ を示すために不等式を使っている. 等式を不等式を使って証明することは解析学における典型的な議論の仕方だと言ってよい. $\\alpha$ と $\\beta$ が等しいことを示すためには, それらの差がどんな正の実数よりも小さいことを示せよばよい. $\\alpha$ と $\\beta$ の差は三角不等式によって, \n", "\n", "$$\n", "|\\beta-\\alpha| =\n", "|(a_n-\\alpha)-(a_n-\\beta)| \\leqq\n", "|a_n-\\alpha| + |a_n-\\beta|\n", "$$\n", "\n", "と評価される. これは $a_n$ と $\\alpha$ の差と $a_n$ と $\\beta$ の差が同時に小さくなれば $\\alpha$ と $\\beta$ の差も小さくなることを意味している. $a_n$ が $\\alpha$ と $\\beta$ に収束するならば, $a_n$ と $\\alpha$ の差も $a_n$ と $\\beta$ の差も同時に小さくできるので, 示したいことが示される. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 収束する実数列 $a_n,b_n$ が全ての番号 $n$ について $a_n \\leqq b_n$ を満たしているならば, $\\ds\\lim_{n\\to\\infty} a_n\\leqq\\lim_{n\\to\\infty}b_n$ が成立することを示せ.\n", "\n", "**解答例1:** $a_n,b_n$ の収束先をそれぞれ $\\alpha,\\beta$ と書くことにする. 一般に実数 $A$ について $-|A|\\leqq A\\leqq |A|$ が成立しているので,\n", "\n", "$$\n", "a_n = a_n-\\alpha+\\alpha \\geqq -|a_n-\\alpha|+\\alpha, \\quad\n", "b_n = b_n-\\beta+\\beta \\leqq |b_n-\\beta|+\\beta\n", "$$\n", "\n", "なので $a_n\\leqq b_n$ なので\n", "\n", "$$\n", "-|a_n-\\alpha|+\\alpha \\leqq a_n\\leqq b_n\\leqq |b_n-\\beta|+\\beta.\n", "$$\n", "\n", "ゆえに\n", "\n", "$$\n", "\\alpha\\leqq \\beta + |a_n-\\alpha| + |b_n-\\beta|.\n", "$$\n", "\n", "$n\\to\\infty$ とすると, $|a_n-\\alpha|\\to 0$, $|b_n-\\beta|\\to 0$ となるので, $\\alpha\\leqq\\beta$ を得る. $\\QED$\n", "\n", "**注意:** 上の証明中で, 全ての番号 $n$ について $a_n0$ を取る. $a_n$ は $\\alpha$ に収束するので, ある番号 $N$ が存在して, $n\\geqq N$ ならば $|a_n-\\alpha|<\\eps$, 特に $\\alpha + \\eps < a_n$ が成立している. $b_n$ は $\\beta$ に収束するので, ある番号 $N'$ が存在して, $n\\geqq N'$ ならば $|b_n-\\beta|<\\eps$, 特に $b_n < \\beta + \\eps$ が成立している. そのとき, $n\\geqq\\max\\{N,N'\\}$ ならば\n", "\n", "$$\n", "\\alpha-\\eps < a_n \\leqq b_n < \\beta + \\eps\n", "$$\n", "\n", "となるので\n", "\n", "$$\n", "\\alpha < \\beta + 2\\eps\n", "$$\n", "\n", "となる.\n", "\n", "$\\eps>0$ は幾らでも小さくできるので, $\\alpha\\leqq\\beta$ が成立している. (もしも $\\alpha>\\beta$ ならば $\\eps = (\\alpha-\\beta)/2 > 0$ のとき $\\alpha < \\beta+2\\eps=\\alpha$ となって矛盾する.) $\\QED$\n", "\n", "**注意:** 収束する実数列 $a_n,b_n$ が全ての番号 $n$ について $a_n < b_n$ を満たしていても, $\\ds\\lim_{n\\to\\infty} a_n < \\lim_{n\\to\\infty}b_n$ が成立するとは限らない. 例えば $a_n=-1/n$, $b_n=1/n$ は $a_n0$ を取る. $a_n$ は $\\alpha$ に収束するので, ある番号 $N'$ が存在して, $n\\geqq N'$ ならば $|a_n-\\alpha|<\\eps/2$ となる. $b_n$ は $\\beta$ に収束するので, ある番号 $N''$ が存在して, $n\\geqq N''$ ならば $|b_n-\\beta|<\\eps/2$ となる. ゆえに, $N=\\max\\{N',N''\\}$ とおくと, $n\\geqq N$ ならば,\n", "\n", "$$\n", "\\begin{aligned}\n", "|(a_n+b_n)-(\\alpha+\\beta)| &= \n", "|(a_n-\\alpha)+(b_n-\\beta)| \n", "\\\\ &\\leqq\n", "|a_n-\\alpha|+|b_n-\\alpha| < \n", "\\frac{\\eps}{2}+\\frac{\\eps}{2} = \\eps.\n", "\\end{aligned}\n", "$$\n", "\n", "ここで $\\leqq$ では三角不等式を使った. これで $a_n+b_n$ が $\\alpha+\\beta$ に収束することが示された. $\\QED$\n", "\n", "**解説:** $a_n+b_n$ が $\\alpha+\\beta$ に収束することを示すためにはそれらの差が適当な条件のもとで適切な意味で小さくなることを示せばよい. それらの差は三角不等式によって\n", "\n", "$$\n", "|(a_n+b_n)-(\\alpha+\\beta)| = \n", "|(a_n-\\alpha)+(b_n-\\beta)| \\leqq\n", "|a_n-\\alpha|+|b_n-\\alpha| \n", "$$\n", "\n", "を満たしている. これは $a_n$ と $\\alpha$ の差と $b_n$ と $\\beta$ の差が両方小さくなれば, $a_n+b_n$ と $\\alpha+\\beta$ の差も小さくなることを意味している. こういう当たり前の議論を $\\eps$-$N$ 論法の言葉で書き直せば上の問題の解答が得られる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** $a_n,b_n$ がそれぞれ $\\alpha,\\beta$ に収束するならば $a_n-b_n$ が $\\alpha-\\beta$ に収束することを示せ. $\\QED$\n", "\n", "上の問題と本質的に同じ証明が可能なので解答は略す." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 数列 $a_n$ がある値に収束しているならば, ある正の実数 $M$ で $|a_n|\\leqq M$ を満たすものが存在することを示せ. (この事実を**収束する数列は有界である**という.)\n", "\n", "**解答例:** $a_n$ の収束先は $\\alpha$ であるとする. ある番号 $N$ が存在して, $n\\geqq N$ ならば $|a_n-\\alpha|\\leqq 1$ となるので, \n", "\n", "$$\n", "|a_n|=|a_n-\\alpha+\\alpha|\\leqq|a_n-\\alpha|+|\\alpha| \\leqq 1+|\\alpha|.\n", "$$\n", "\n", "ゆえに $M=\\max\\{|a_1|,\\ldots,|a_{N-1}|,1+|\\alpha|\\}$ とおくと, すべての番号 $n$ について $|a_n|\\leqq M$ となる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**例:** $\\ds a_n=10+\\frac{1000}{n}$ である場合に上の解答例の議論を適用してみよう. そのとき, $n\\geqq 1000$ とすると, $\\ds|a_n-10|=\\frac{1000}{n}\\leqq \\frac{10}{1000}\\leqq 1$ となり, $|a_n|\\leqq 1+10=11$ となる. $a_1=1010$, $a_2=510$, $\\ldots$, $a_{999}=10+1000/999$, $1+10=11$ の最大値は $1010$ である. ゆえにすべての番号 $n$ について $|a_n|\\leqq 1010$. $\\QED$ " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 数列 $a_n,b_n$ がそれぞれ $\\alpha,\\beta$ に収束するとき, 数列 $a_n b_n$ が $\\alpha\\beta$ に収束することを示せ.\n", "\n", "**解答例1:** 収束する数列は有界なので, ある正の実数 $M$ ですべての番号 $n$ について $|a_n|\\leqq M$ を満たすものが存在する. 三角不等式より, $n\\to\\infty$ のとき\n", "\n", "$$\n", "\\begin{aligned}\n", "|a_n b_n - \\alpha\\beta| &= \n", "|a_n b_n - a_n\\beta + a_n\\beta - \\alpha\\beta| =\n", "|a_n(b_n - \\beta) + (a_n-\\alpha)\\beta|\n", "\\\\ &\\leqq \n", "|a_n||b_n-\\beta|+|a_n-\\alpha||\\beta| \\leqq \n", "M|b_n-\\beta|+|a_n-\\alpha||\\beta| \\to 0\n", "\\end{aligned}\n", "$$\n", "\n", "となるので, $a_n b_n \\to \\alpha\\beta$ となる. $\\QED$\n", "\n", "**解答例2:** $a_n b_n$ と $\\alpha\\beta$ の差は三角不等式によって\n", "\n", "$$\n", "\\begin{aligned}\n", "|a_n b_n - \\alpha\\beta| &= \n", "|a_n b_n - a_n\\beta + a_n\\beta - \\alpha\\beta| =\n", "|a_n(b_n - \\beta) + (a_n-\\alpha)\\beta|\n", "\\\\ &\\leqq \n", "|a_n||b_n-\\beta|+|a_n-\\alpha||\\beta|\n", "\\end{aligned}\n", "$$\n", "\n", "と上から評価される. 上の問題の結果より, ある正の実数 $M$ が存在して, すべての番号 $n$ について $|a_n|\\leqq M$ となる. $b_n$ は $\\beta$ に収束するので, ある番号 $N''$ が存在して, $n\\geqq N''$ ならば $|b_n-\\beta|<\\eps/(2M)$ となる. $a_n$ は $\\alpha$ に収束するので, ある番号 $N'$ が存在して, $n\\geqq N'$ ならば $|a_n-\\alpha|<\\eps/(2|\\beta|+1)$ となる. そのとき, $n\\geqq\\max\\{N',N''\\}$ ならば, 上の評価式より, \n", "\n", "$$\n", "|a_n b_n - \\alpha\\beta| < M\\frac{\\eps}{2M}+\\frac{\\eps}{2|\\beta|+1}|\\beta| \\leqq\n", "\\frac{\\eps}{2}+\\frac{\\eps}{2} = \\eps.\n", "$$\n", "\n", "これで $a_n b_n$ が $\\alpha\\beta$ に収束することが示された. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** $a_n$ が $\\alpha\\ne 0$ に収束するならば, $r<|\\alpha|$ を満たす任意の実数 $r$ に対して, ある番号 $N$ が存在して, $n\\geqq N$ ならば $|a_n|\\geqq r$ となることを示せ.\n", "\n", "**解答例:** $r<|\\alpha|$ より $\\eps=|\\alpha|-r$ とおくと, $\\eps>0$ となり, $a_n$ は $\\alpha$ に収束するので, ある番号 $N$ が存在して, $n\\geqq N$ ならば, $|a_n-\\alpha|<\\eps=|\\alpha|-r$ となり, \n", "\n", "$$\n", "|a_n| = |a_n-\\alpha+\\alpha| \\geqq |\\alpha|-|a_n-\\alpha| > r\n", "$$\n", "\n", "となる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**例:** $\\ds a_n = -1 + (-1)^n \\frac{100}{n}$ と $r=0.9$ の場合に上の解答例の議論を適用してみよう. $n\\geqq 1001$ とすると $\\ds|a_n-(-1)|=\\frac{100}{n}\\leqq \\frac{100}{1001} < 0.1$ となるので, $\\ds |a_n|\\geqq|-1| - |a_n-(-1)| > 1-0.1=0.9=r$ となる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** $a_n$ が $\\alpha\\ne 0$ に収束しているならば $1/a_n$ は $1/\\alpha$ に収束することを示せ.\n", "\n", "**解答例1:** $a_n\\to\\alpha\\ne 0$ より, 十分に $n$ を大きくすると $\\ds|a_n|\\geqq\\frac{|\\alpha|}{2}$ となる. 三角不等式より, $n\\to\\infty$ のとき, \n", "\n", "$$\n", "\\left|\\frac{1}{a_n}-\\frac{1}{\\alpha}\\right| =\n", "\\left|\\frac{\\alpha-a_n}{a_n \\alpha}\\right| =\n", "\\frac{|a_n-\\alpha|}{|a_n||\\alpha|}\\leqq\n", "\\frac{|a_n-\\alpha|}{(|\\alpha|/2)|\\alpha|}\\to 0.\n", "$$\n", "\n", "ゆえに $\\ds\\frac{1}{a_n}\\to\\frac{1}{\\alpha}$ となる. $\\QED$\n", "\n", "**解答例2:** $1/a_n$ と $1/\\alpha$ の差は\n", "\n", "$$\n", "\\left|\\frac{1}{a_n}-\\frac{1}{\\alpha}\\right| =\n", "\\left|\\frac{\\alpha-a_n}{a_n \\alpha}\\right| =\n", "\\frac{|a_n-\\alpha|}{|\\alpha|}\n", "$$\n", "\n", "をみたしている. 上の問題の結果より, ある番号 $N'$ が存在して, $n\\geqq N'$ ならば $|a_n|\\geqq |\\alpha|/2$ となる. $a_n$ は $\\alpha$ に収束しているので, ある番号 $N''$ が存在して, $n\\geqq N''$ ならば $|a_n-\\alpha|<|\\alpha|^2\\eps/2$ となる. そのとき, $N=\\max\\{N',N''\\}$ とおくと, $n\\geqq N$ ならば\n", "\n", "$$\n", "\\left|\\frac{1}{a_n}-\\frac{1}{\\alpha}\\right| =\n", "\\frac{|a_n-\\alpha|}{|a_n||\\alpha|} < \\frac{|\\alpha|^2\\eps/2}{(|\\alpha|/2)|\\alpha|} = \\eps.\n", "$$\n", "\n", "これで $1/a_n$ が $\\alpha$ に収束することが示された. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 数列 $a_n$ が $\\alpha$ に収束しているならば\n", "\n", "$$\n", "b_n = \\frac{a_1+\\cdots+a_n}{n}\n", "$$\n", "\n", "も $\\alpha$ に収束することを示せ.\n", "\n", "**ヒント:** $a_n$ を $a_n-\\alpha$ で置き換えることによって, $a_n$ が $0$ で収束するならば $b_n$ も $0$ に収束することを示せば十分であることがわかる. $a_n$ が $0$ に収束するならば, 十分大きな $N$ について $a_n$ はほとんど $0$ になるので, $n\\geqq N$ のとき加法平均 $(a_N+a_{N+1}+\\cdots+a_n)/(n-N+1)$ もほとんど $0$ になるだろう. $N$ を固定したまま $n\\geqq N$ を大きくすれば $(a_1+\\cdots+a_{N-1})/n$ は $0$ に収束する. $\\QED$\n", "\n", "**注意:** 上の問題は $\\eps$-$N$ 論法を使った方が「易しい」例として有名である. $\\eps$-$N$ 論法を理解しているかどうかのチェックにもよく使われる. 以下の解答例を読まずに自力で証明を考えた方がよいと思われる. $\\QED$\n", "\n", "**解答例1:** $A_n=a_n-\\alpha$ とおくと, \n", "\n", "$$\n", "b_n - \\alpha = \n", "\\frac{a_1+\\cdots+a_n}{n} -\\alpha =\n", "\\frac{(a_1-\\alpha)+\\cdots+(a_n-\\alpha)}{n} =\n", "\\frac{A_1+\\cdots+A_n}{n}.\n", "$$\n", "\n", "これの右辺を $B_n$ を書くことにする. これより, $A_n\\to 0$ から $B_n\\to 0$ を示せば十分であることがわかる. 以下では $A_n$, $B_n$ をそれぞれ $a_n$, $b_n$ と書く. $a_n\\to 0$ という仮定のもとで $b_n\\to 0$ を示せばよい.\n", "\n", "任意に $\\eps>0$ を取る. $a_n\\to 0$ より, ある番号 $N$ が存在して, $|a_N|,|a_{N+1}|,|a_{N+2}|,\\ldots$ がすべて $\\ds\\frac{\\eps}{2}$ 未満になるので, $n\\geqq N$ ならば\n", "\n", "$$\n", "\\begin{aligned}\n", "|b_n|&\\leqq \\frac{|a_1|+\\cdots+|a_n|}{n} \n", "\\\\ &=\n", "\\frac{|a_1|+\\cdots+|a_{N-1}|}{n} + \\frac{|a_N|+|a_{N+1}|+\\cdots+|a_n|}{n}\n", "\\\\ &\\leqq\n", "\\frac{|a_1|+\\cdots+|a_{N-1}|}{n} + \\frac{|a_N|+|a_{N+1}|+\\cdots+|a_n|}{n-N+1}\n", "\\\\ &<\n", "\\frac{|a_1|+\\cdots+|a_{N-1}|}{n} + \\frac{\\eps}{2}.\n", "\\end{aligned}\n", "$$\n", "\n", "$N'$ は $2(|a_1|+\\cdots+|a_{N-1}|)/\\eps$ より大きな正の整数であるとする. そのとき, $n\\geqq N'$ ならば\n", "\n", "$$\n", "\\frac{|a_1|+\\cdots+|a_{N-1}|}{n} < \\frac{|a_1|+\\cdots+|a_{N-1}|}{2(|a_1|+\\cdots+|a_{N-1}|)/\\eps} = \n", "\\frac{\\eps}{2}.\n", "$$\n", "\n", "以上より, $n\\geqq \\max\\{N,N'\\}$ ならば\n", "\n", "$$\n", "|b_n| < \\frac{\\eps}{2}+\\frac{\\eps}{2} = \\eps.\n", "$$\n", "\n", "これで $b_n$ が $0$ に収束することが示された. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**注意:** 上の問題の逆は成立しない. すなわち $\\ds b_n = \\frac{a_1+\\cdots+a_n}{n}$ が収束していても, $a_n$ が収束しない場合がある. 例えば, $a_n=(-1)^n$ のとき, $a_n$ は振動して収束しないが, $b_n$ は $0$ に収束する. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 数列 $a_n$ が収束するならば $a_{n+1}-a_n$ も $0$ に収束することを示せ.\n", "\n", "**解答例1:** 数列 $a_n$ は $\\alpha$ に収束していると仮定する. 三角不等式より, $n\\to\\infty$ のとき\n", "\n", "$$\n", "|a_{n+1}-a_n| = |a_{n+1}-\\alpha+\\alpha-a_n|\\leqq\n", "|a_{n+1}-\\alpha|+|\\alpha-a_n| \\to 0.\n", "$$\n", "\n", "これで, $a_{n+1}-a_n$ が $0$ に収束することが示された. $\\QED$\n", "\n", "**解答例2:** 数列 $a_n$ は $\\alpha$ に収束すると仮定し, $\\eps>0$ を任意に取る. $a_n$ が $\\alpha$ に収束することから, ある番号 $N$ が存在して, $n\\geqq N$ ならば $|a_n-\\alpha|<\\eps/2$ となる. そのとき, $n\\geqq N$ ならば\n", "\n", "$$\n", "|a_{n+1}-a_n| = |a_{n+1}-\\alpha+\\alpha-a_n|\\leqq\n", "|a_{n+1}-\\alpha|+|\\alpha-a_n| < \\frac{\\eps}{2}+\\frac{\\eps}{2} = \\eps.\n", "$$\n", "\n", "これで $a_{n+1}-a_n$ が $0$ に収束することが示された. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**注意:** 上の問題の逆は成立しない. すなわち $a_{n+1}-a_n\\to 0$ であっても $a_n$ が収束しない場合がある. 例えば, $\\ds a_n=\\frac{1}{1}+\\frac{1}{2}+\\cdots+\\frac{1}{n}$ のとき, $\\ds a_{n+1}-a_n=\\frac{1}{n+1}$ は $0$ に収束するが, $a_n$ は無限大に発散する. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題(Cauchy列=基本列):** 数列 $a_n$ が**Cauchy列**もしくは**基本列**であるとは, 任意の $\\eps>0$ に対して, ある番号 $N$ が存在して, $m,n\\geqq N$ ならば $|a_m-a_n|\\leqq \\eps$ となることだと定める. 収束する数列はCauchy列であることを示せ.\n", "\n", "**解答例:** 本質的に上の問題の解答例と同じ. 数列 $a_n$ は $\\alpha$ に収束すると仮定し, $\\eps>0$ を任意に取る. $a_n$ が $\\alpha$ に収束することから, ある番号 $N$ が存在して, $n\\geqq N$ ならば $|a_n-\\alpha|<\\eps/2$ となる. そのとき, $m,n\\geqq N$ ならば\n", "\n", "$$\n", "|a_m-a_n| = |a_m-\\alpha+\\alpha-a_n|\\leqq\n", "|a_m-\\alpha|+|\\alpha-a_n| < \\frac{\\eps}{2}+\\frac{\\eps}{2} = \\eps.\n", "$$\n", "\n", "これで示すべきことが全て示された. $\\QED$\n", "\n", "**注意1:** 数列 $a_n$ がCauchy列であることを, 「$m,n\\to\\infty$ のとき $|a_m-a_n|\\to 0$ になる」と言うこともある. その代わりに「$m, n\\geqq N$ で $N\\to\\infty$ のとき $|a_m-a_n|\\to 0$ となる」と言う方がよいかもしれない. $\\QED$\n", "\n", "**注意2(完備性の定義):** $a_n$ が収束する数列ならば $a_n$ はCauchy列になる. 実数や複素数の範囲内ではこれの逆が成立している. すなわち, $a_n$ が実数列(もしくは複素数列)でかつCauchy列ならば, ある実数 $\\alpha$ (もしくは複素数 $\\alpha$)が存在して $a_n$ は $\\alpha$ に収束する. この結果を「実数(もしくは複素数)全体の集合は**完備**である」という.\n", "\n", "実数全体の集合は有理数全体の集合の通常の絶対値による距離に関する**完備化**として定義されていると思ってよい. その意味で実数全体の集合の定義の中には完備性が含まれていると思ってよい. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 級数の収束の定義\n", "\n", "(無限)級数 $\\ds\\sum_{n=1}^\\infty a_n$ の収束は数列 $\\ds s_n = \\sum_{k=1}^n a_k$ の収束で定義される. 収束するとき $\\ds \\sum_{n=1}^\\infty a_n = \\lim_{n\\to\\infty} \\sum_{k=1}^n a_k$ と書く." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**例(等比級数):** $|q|<1$, $a_n=q^{n-1}$ のとき\n", "\n", "$$\n", "s_n = \\sum_{k=1}^n a_k = 1+q+q^2+\\cdots+q^{n-1} = \\frac{1-q^n}{1-q}\n", "$$\n", "\n", "であり, $n\\to\\infty$ で $q^n\\to 0$ なので, $s_n\\to 1/(1-q)$ となる. すなわち,\n", "\n", "$$\n", "\\sum_{m=0}^\\infty q^m = \\sum_{n=1}^\\infty q^{n-1} = \\frac{1}{1-q}.\n", "\\qquad \\QED\n", "$$\n", "\n", "**注意:** $\\ds1+q+\\cdots+q^{n-1}=\\frac{1-q^n}{1-q}$ は $q\\to 1$ で $n$ に収束する. これを $q$ 数 ($q$-number)と呼び, $(n)_q$ のように書くことがある:\n", "\n", "$$\n", "(n)_q = \\frac{1-q^n}{1-q}.\n", "$$\n", "\n", "さらに $q$ 階乗 ($q$-factorial)や $q$ 二項係数が\n", "\n", "$$\n", "(n)_q! = (1)_q(2)_q\\cdots(n)_q, \\quad\n", "\\binom{n}{k}_q = \\frac{(n)_q!}{(k)_q!\\,(n-k)_q!}\n", "$$\n", "\n", "と定義される. このようにして, 数学的対象にパラメーター $q$ を入れて解析を行うことを $q$ 解析と呼ぶ. 20世紀の終わり頃に量子群が発見され, $q$ 解析と深い繋がりがあることが判明した. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 級数 $\\ds\\sum_{n=1}^\\infty a_n$ が収束しているならば, $n\\to\\infty$ で $a_n\\to 0$ となることを示せ.\n", "\n", "**解答例1:** 級数 $\\ds\\sum_{n=1}^\\infty a_n$ は $\\alpha$ に収束していると仮定する. すなわち, $\\ds \\sum_{k=1}^n a_k$ は $n\\to\\infty$ で $\\alpha$ に収束していると仮定する. このとき, $n\\to\\infty$ のとき,\n", "\n", "$$\n", "|a_n| = \\left|\\sum_{k=1}^n a_k - \\sum_{k=1}^{n-1}a_k\\right| =\n", "\\left|\\sum_{k=1}^n a_k-\\alpha\\right|+\\left|\\sum_{k=1}^{n-1}a_k-\\alpha\\right| \\to 0.\n", "$$\n", "\n", "これで $a_n\\to 0$ であることが示された. $\\QED$\n", "\n", "**解答例2:** 級数 $\\ds\\sum_{n=1}^\\infty a_n$ は $\\alpha$ に収束していると仮定し, $\\eps >0$ を任意に取る. このとき, ある番号 $N$ が存在して, $n\\geqq N$ ならば $\\ds\\left|\\sum_{k=1}^n a_k - \\alpha\\right|<\\frac{\\eps}{2}$ となる. ゆえに $n-1\\geqq N$ のとき, \n", "\n", "$$\n", "|a_n| = \\left|\\sum_{k=1}^n a_k - \\sum_{k=1}^{n-1}a_k\\right| =\n", "\\left|\\sum_{k=1}^n a_k-\\alpha\\right|+\\left|\\sum_{k=1}^{n-1}a_k-\\alpha\\right| <\n", "\\frac{\\eps}{2} + \\frac{\\eps}{2} = \\eps.\n", "$$\n", "\n", "これで, $a_n\\to 0$ となることが示された. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 上の問題の逆が成立しないことを示せ. すなわち, $a_n\\to 0$ であっても, $\\ds\\sum_{n=1}^\\infty a_n$ が収束するとは限らないことを示せ.\n", "\n", "**解答例:** $a_n=1/n$ とおくと, $a_n\\to 0$ である. しかし, $n\\leqq x\\leqq n+1$ のとき $1/x\\leqq 1/n$ なので, \n", "\n", "$$\n", "\\log(k+1)-\\log k = \\int_k^{k+1}\\frac{dx}{x} \\leqq \\int_k^{k+1}\\frac{1}{k}\\,dx = \\frac{1}{k} = a_k\n", "$$\n", "\n", "となるので, 両辺を $k=1,\\ldots,n$ について足し上げてると, \n", "\n", "$$\n", "\\log(n+1) \\leqq \\sum_{k=1}^n \\frac{1}{k} = \\sum_{k=1}^n a_k.\n", "$$\n", "\n", "これより, $n\\to\\infty$ で $\\ds \\sum_{k=1}^n a_k\\to\\infty$ となることがわかる. $\\QED$" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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"image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 1/1+1/2+...+1/n と log n の比較\n", "# nを大きくするとそれらの差はEuler定数 γ = 0.5772… に近付く\n", "# ゆえに, 1/1+1/2+...+1/n は log n + γ でよく近似される.\n", "# 以下のプロットでもそれらのグラフはほとんど一致している.\n", "\n", "f(x) = sum(k->1/k, 1:floor(Int, x))\n", "n = 1:100\n", "x = 1:0.1:100\n", "plot(size=(500,350), legend=:bottomright)\n", "plot!(n, f.(n), label=\"1/1+1/2+...+1/n\", lw=2.5)\n", "plot!(x, log.(x), label=\"log(x)\", ls=:dashdot, color=:orange)\n", "plot!(x, log.(x.+1), label=\"log(x+1)\", ls=:dashdot, color=:darkgreen)\n", "plot!(x, log.(x).+eulergamma, label=\"log(x) + Euler's gamma\", lw=2, ls=:dash, color=:red)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "search: \u001b[0m\u001b[1me\u001b[22m\u001b[0m\u001b[1mu\u001b[22m\u001b[0m\u001b[1ml\u001b[22m\u001b[0m\u001b[1me\u001b[22m\u001b[0m\u001b[1mr\u001b[22m\u001b[0m\u001b[1mg\u001b[22m\u001b[0m\u001b[1ma\u001b[22m\u001b[0m\u001b[1mm\u001b[22m\u001b[0m\u001b[1mm\u001b[22m\u001b[0m\u001b[1ma\u001b[22m\n", "\n" ] }, { "data": { "text/latex": [ "\\begin{verbatim}\n", "γ\n", "eulergamma\n", "\\end{verbatim}\n", "Euler's constant.\n", "\n", "\\section{Examples}\n", "\\begin{verbatim}\n", "julia> Base.MathConstants.eulergamma\n", "γ = 0.5772156649015...\n", "\n", "julia> dx = 10^-6;\n", "\n", "julia> sum(-exp(-x) * log(x) for x in dx:dx:100) * dx\n", "0.5772078382499133\n", "\\end{verbatim}\n" ], "text/markdown": [ "```\n", "γ\n", "eulergamma\n", "```\n", "\n", "Euler's constant.\n", "\n", "# Examples\n", "\n", "```jldoctest\n", "julia> Base.MathConstants.eulergamma\n", "γ = 0.5772156649015...\n", "\n", "julia> dx = 10^-6;\n", "\n", "julia> sum(-exp(-x) * log(x) for x in dx:dx:100) * dx\n", "0.5772078382499133\n", "```\n" ], "text/plain": [ "\u001b[36m γ\u001b[39m\n", "\u001b[36m eulergamma\u001b[39m\n", "\n", " Euler's constant.\n", "\n", "\u001b[1m Examples\u001b[22m\n", "\u001b[1m ≡≡≡≡≡≡≡≡≡≡\u001b[22m\n", "\n", "\u001b[36m julia> Base.MathConstants.eulergamma\u001b[39m\n", "\u001b[36m γ = 0.5772156649015...\u001b[39m\n", "\u001b[36m \u001b[39m\n", "\u001b[36m julia> dx = 10^-6;\u001b[39m\n", "\u001b[36m \u001b[39m\n", "\u001b[36m julia> sum(-exp(-x) * log(x) for x in dx:dx:100) * dx\u001b[39m\n", "\u001b[36m 0.5772078382499133\u001b[39m" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "?eulergamma" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 連続的な極限\n", "\n", "函数の $x$ における値 $f(x)$ が $x\\to a$ で $\\alpha$ に**収束する**とは, 任意の $\\eps>0$ に対して, ある $\\delta > 0$ が存在して, $0<|x-a|<\\delta$ を満たすすべての $x$ について $|f(x)-\\alpha|<\\eps$ が成立することである.\n", "\n", "$f(x)$ が $x\\to a$ で $\\alpha$ に収束するとき, その極限の値を $\\displaystyle \\lim_{x\\to a} f(x) = \\alpha$ と定義する. 収束しないときには極限の値は定義されない." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 上の定義における $|f(x)-\\alpha|<\\eps$ を $|f(x)-\\alpha|\\leqq\\eps$ に置き換えても同値な条件になることを示せ. $\\QED$\n", "\n", "**解答例:** 以下の2条件が互いに同値であることを示そう.\n", "\n", "(1) 任意の $\\eps>0$ に対して, ある $\\delta > 0$ が存在して, $0<|x-a|<\\delta$ を満たすすべての $x$ について $|f(x)-\\alpha|<\\eps$ が成立する.\n", "\n", "(2) 任意の $\\eps>0$ に対して, ある $\\delta > 0$ が存在して, $0<|x-a|<\\delta$ を満たすすべての $x$ について $|f(x)-\\alpha|\\leqq \\eps$ が成立する.\n", "\n", "ついでに任意の正の実数 $M$ について以下も同値であることも示してしまおう.\n", "\n", "(3) 任意の $\\eps>0$ に対して, ある $\\delta > 0$ が存在して, $0<|x-a|<\\delta$ を満たすすべての $x$ について $|f(x)-\\alpha|\\leqq M\\eps$ が成立する.\n", "\n", "(1) $\\implies$ (2): (1)を仮定して, 任意に $\\eps>0$ を取る. (1)より, $0<|x-a|<\\delta$ を満たすすべての $x$ について $|f(x)-\\alpha|< \\eps$ が成立し, そのとき $|f(x)-\\alpha|\\leqq\\eps$ が自明に成立している. これで(1)から(2)が自明に導かれることがわかった.\n", "\n", "(2) $\\implies$ (3): (2)を仮定して, 任意に $\\eps>0$ を取る. (2)の $\\eps$ として $M\\eps$ を採用すると, ある $\\delta > 0$ が存在して, $0<|x-a|<\\delta$ を満たすすべての $x$ について $|f(x)-\\alpha|\\leqq M\\eps$ が成立する. これで(2)から(3)も自明に導かれることがわかった.\n", "\n", "(3) $\\implies$ (1): (3)を仮定して, 任意に $\\eps>0$ を取る. (3)の $\\eps$ として $\\eps/(2M)$ を採用すると, ある ある $\\delta > 0$ が存在して, $0<|x-a|<\\delta$ を満たすすべての $x$ について $|f(x)-\\alpha|\\leqq M(\\eps/(2M))=\\eps/2<\\eps$ が成立する. これで(3)から(1)が導かれることがわかった.\n", "\n", "以上によって(1),(2),(3)が互いに同値であることがわかった. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** さらに $<\\delta$ の部分を $\\leqq\\delta$ に変更しても同値であることを示せ. \\QED\n", "\n", "上の解答例と完全に同じ考え方で証明できるので解答略." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 収束の判定法の基本" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 挟撃法 (挟み撃ち)\n", "\n", "**定理:** $A_n \\leqq a_n \\leqq B_n$ でかつ $A_n$ と $B_n$ が $\\alpha$ に収束するならば $a_n$ も $\\alpha$ に収束する. \n", "\n", "**証明:** $\\eps>0$ を任意に取る. $A_n\\to\\alpha$ より, ある番号 $N'$ が存在して, $n\\geqq N'$ ならば $\\ds|A_n-\\alpha|<\\eps$, 特に $\\ds\\alpha-\\eps0$ が存在して, $0<|x|<\\delta$ ならば $f(x)\\geqq M$ となることである. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 二項定理の復習\n", "\n", "以下の節では $0$ への収束の判定を主に扱うが, そのための準備として二項定理について復習しておく.\n", "\n", "二項係数を\n", "\n", "$$\n", "\\binom{n}{k} = \\frac{n(n-1)\\cdots(n-k+1)}{k!}\n", "$$\n", "\n", "と定義する. 例えば\n", "\n", "$$\n", "\\binom{n}{0} = 1, \\quad\n", "\\binom{n}{1} = n, \\quad\n", "\\binom{n}{2} = \\frac{n(n-1)}{2}, \\quad\n", "\\binom{n}{3} = \\frac{n(n-1)(n-2)}{6}.\n", "$$\n", "\n", "$n$ も $k$ も非負の整数のとき $\\binom{n}{k}$ は $n$ 個から $k$ 個選び出すときの場合の数に一致している. 高校数学では ${}_nC_k$ のように書いているようだが, 一般には $\\binom{n}{k}$ と書くことの方が多いように感じられるので, $\\binom{n}{k}$ の方を使用する. (上の定義に従えば, $k$ が非負の整数でありさえすれば, $n$ が整数でなくても $\\binom{n}{k}$ が定義されていることに注意せよ. この事実は二項展開を考えるときに必要になる.)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "以上の記号法のもとで, 二項定理は\n", "\n", "$$\n", "(x+y)^n = \n", "\\sum_{k=0}^n \\binom{n}{k} x^k y^{n-k} =\n", "\\sum_{k=0}^n \\binom{n}{k} x^{n-k}y^k\n", "\\tag{$*$}\n", "$$\n", "\n", "と書ける. ２つ目の等号は自明である.\n", "\n", "高校数学で証明を習っているはずだが, 忘れている人は二項定理を証明しておくこと. \n", "\n", "**考え方:** 解析学では不等式を扱いたい. しかし, 二項定理のような**良い等式は良い不等式を生み出す**ために非常に役に立つ. 解析学では非自明な等式は非自明な不等式を作る材料になる. 次の節の証明を見よ!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**二項定理の証明1:** $(x+y)^n$ を展開したときに得られる $x^k y^{n-k}$ の係数は $k$ 個の $x$ と $n-k$ 個の $y$ の並べ方の個数に等しい. その個数は全部で $n$ 個並べるうち $k$ 個の $x$ の場所の選び方の個数に等しい. それは\n", "\n", "$$\n", "\\frac{n!}{k!(n-k)!} = \\binom{n}{k}\n", "$$\n", "\n", "に等しい. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**二項定理の証明2:** $n$ に関する帰納法を使う. $n=0$ の場合に($*$)は成立している. ($*$)が $n$ について成立していると仮定する. そのとき, $k$ が負の整数のとき $\\binom{n}{k}=0$ と約束しておくと, \n", "\n", "$$\n", "\\begin{aligned}\n", "\\binom{n}{k-1} + \\binom{n}{k} &= \n", "\\frac{n(n-1)\\cdots(n-k+2)}{(k-1)!} + \\frac{n(n-1)\\cdots(n-k+2)(n-k+1)}{k!}\n", "\\\\ &=\n", "\\frac{n(n-1)\\cdots(n-k+2)}{k!}\\times(k+(n-k+1))\n", "\\\\ &=\n", "\\frac{n(n-1)\\cdots(n-k+2)}{k!}\\times(n+1)\n", "\\\\ &=\n", "\\binom{n+1}{k}.\n", "\\end{aligned}\n", "$$\n", "\n", "ゆえに $n$ に関する帰納法の仮定($*$)を使うと, \n", "\n", "$$\n", "\\begin{aligned}\n", "(x+y)^{n+1} &= (x+y)(x+y)^n =\n", "(x+y)\\sum_{k=0}^n \\binom{n}{k}x^k y^{n-k}\n", "\\\\ &=\n", "\\sum_{k=0}^n \\binom{n}{k}x^{k+1}y^{n-k} +\n", "\\sum_{k=0}^n \\binom{n}{k}x^{k}y^{n+1-k}\n", "\\\\ &=\n", "\\sum_{k=0}^{n+1} \\binom{n}{k-1}x^{k}y^{n+1-k} +\n", "\\sum_{k=0}^{n+1} \\binom{n}{k} x^{k}y^{n+1-k}\n", "\\\\ &=\n", "\\sum_{k=0}^{n+1} \\left(\\binom{n}{k-1}+\\binom{n}{k}\\right)x^{k}y^{n+1-k}\n", "\\\\ &=\n", "\\sum_{k=0}^{n+1} \\binom{n+1}{k}x^{k}y^{n+1-k}.\n", "\\end{aligned}\n", "$$\n", "\n", "これで $n$ を $n+1$ で置き換えた場合の条件($*$)も成立することがわかった. これで示すべきことが示された. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**注意:** 二項定理は形式べき級数としての $\\ds f(x)=\\sum_{k=0}^\\infty\\frac{x^k}{k!}$ が\n", "\n", "$$\n", "f(x+y) = f(x)f(y)\n", "$$\n", "\n", "を満たしていることと同値である. なぜならば,\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "f(x+y) = \\sum_{n=0}^\\infty\\frac{1}{n!}(x+y)^n,\n", "\\\\ &\n", "f(x)f(y) = \n", "\\sum_{k,l=0}^\\infty \\frac{x^k y^l}{k!l!} =\n", "\\sum_{n=0}^\\infty \\sum_{k=0}^{n-k} \\frac{x^k y^{n-k}}{k!(n-k)!} =\n", "\\sum_{n=0}^\\infty \\frac{1}{n!}\\sum_{k=0}^{n-k} \\binom{n}{k} x^k y^{n-k}.\n", "\\end{aligned}\n", "$$\n", "\n", "$f(x)$ は $e^{x+y}=e^x e^y$ を満たす指数函数 $e^x$ の $x=0$ におけるTaylor展開に等しい. そのことを使うことを許せば, 二項定理を経由せずに $f(x+y)=f(x)f(y)$ を示せる. このように, 二項定理のような組み合わせ論的な内容を持つ定理と $e^{x+y}=e^x e^y$ のような特別な函数が満たす特別な関係式が同値になっていることはよくある. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 多項式函数より指数函数の方が速く増加すること\n", "\n", "**定理:** $a>1$ ならば $n\\to\\infty$ のとき $\\ds\\frac{n^l}{a^n}\\to 0$. $\\QED$\n", "\n", "**証明:** 二項定理を使う. \n", "\n", "$L$ は $l$ より大きな正の整数であるとし, $n$ は $L$ 以上であると仮定する. \n", "\n", "そのとき特に $n\\to\\infty$ のとき $n^l/n^L\\to 0$ が成立している.\n", "\n", "$a>1$ なので $a=1+\\alpha$, $\\alpha>0$ と書ける. 二項定理より\n", "\n", "$$\n", "\\begin{aligned}\n", "a^n &= (1+\\alpha)^n = \\sum_{k=0}^n \\binom{n}{k}\\alpha^k =\n", "\\sum_{k=0}^n \\frac{n(n-1)\\cdots(n-k+1)}{k!}\\alpha^2 \n", "\\\\ &\\geqq\n", "\\frac{n(n-1)\\cdots(n-L+1)}{L!}\\alpha^L\n", "= n^L\\frac{(1-1/n)\\cdots(1-(L-1)/n)\\alpha^L}{L!}.\n", "\\end{aligned}\n", "$$\n", "\n", "ゆえに, $n\\to\\infty$ のとき\n", "\n", "$$\n", "0\\leqq\\frac{n^l}{a^n} \\leqq \n", "\\frac{n^l}{n^L}\\times\\frac{L!}{(1-1/n)\\cdots(1-(L-1)/n)\\alpha^L} \\to\n", "0\\times\\frac{L!}{\\alpha^L} = 0.\n", "\\qquad\\QED\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** $a=1.01$, $l=100$ のとき, $n^l$ と $a^n$ の大きさを比較するためのグラフを描いてみよ.\n", "\n", "**解答例:** 直接 $n^l$ と $a^n$ を扱うと数値が巨大になり過ぎるので, それらの対数を比較することにする." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = 1.01\n", "l = 100\n", "n = 1:10^3:2*10^5\n", "f(n) = l*log(n)\n", "g(n) = n*log(a)\n", "plot(size=(500, 350), legend=:topleft, xlabel=\"n\")\n", "plot!(n, f.(n), label=\"$l*log(n)\")\n", "plot!(n, g.(n), label=\"n*log($a)\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** $a>1$ ならば, $x>0$ が連続的に幾らでも大きくなるとき $\\ds\\frac{x^l}{a^x}\\to 0$ となることを示せ.\n", "\n", "**証明:** 正の整数 $n$ について $n\\leqq x \\leqq n+1$ のとき, $x^l$ は $n^l$ と $(n+1)^l$ のあいだにあり, $a^x$ は $a^n$ 以上になる. ゆえに, $x\\to\\infty$ のとき, $n\\to\\infty$ となり, \n", "\n", "$$\n", "0\\leqq\n", "\\frac{x^l}{a^x} \\leqq \\frac{\\max\\{n^l, (n+1)^l\\}}{a^n} \\leqq \n", "\\max\\left\\{ \\frac{n^l}{a^n},\\; a\\frac{(n+1)^l}{a^{n+1}}\\right\\} \\to 0.\n", "$$\n", "\n", "最後に上の方の定理を使った. これで示すべきことが示された. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** $x>0$ について, $x\\to 0$ のとき $x\\log x\\to 0$ となることを示せ.\n", "\n", "**証明:** $x>0$ なので $x=e^{-t}$, $t\\in\\R$ と書ける. $x\\to 0$ のとき $t\\to\\infty$ となるので,\n", "\n", "$$\n", "x\\log x = e^{-t}\\log e^{-t} = -\\frac{t}{e^t} \\to 0.\n", "$$\n", "\n", "最後の上の問題の結果を使った. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**注意:** 統計学に出て来る計算では $0\\log 0 = 0$ と約束しておくことが適切な場合が多い. $\\QED$" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(x) = x*log(x)\n", "x = 0.001:0.001:1\n", "plot(x, f.(x), xlim=(-0.1,1), ylim=(-0.4,0), label=\"y = x log x\", legend=:top)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 指数函数より階乗の方が速く増加すること" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**定理:** $a>0$ と仮定する. そのとき $n\\to\\infty$ ならば $\\ds\\frac{a^n}{n!}\\to 0$ が成立する.\n", "\n", "**証明:** $N$ を十分大きくすると $\\ds\\frac{a}{N+1}<1$ となる. そして, $n>N$ とすると, \n", "\n", "$$\n", "\\frac{a^n}{n!} = \\frac{a^N}{N!}\\frac{a}{N+1}\\frac{a}{N+2}\\cdots\\frac{a}{n}\n", "\\leqq \\frac{a^N}{N!}\\left(\\frac{a}{N+1}\\right)^{n-N} \\to 0 \\quad(n\\to\\infty).\n", "\\quad \\QED\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** $a>1$ と仮定する. そのとき $n\\to\\infty$ ならば $\\ds\\frac{n!}{a^{n^2}}\\to 0$ が成立することを示せ.\n", "\n", "**解答例:** 指数函数は多項式函数より速く増加するので $\\ds\\frac{n}{a^n}\\to 0$. ゆえにある番号 $N$ が存在して $n>N$ ならば $\\frac{n}{a^n} \\geqq \\frac{1}{2}$ となる. したがって, $n>N$ のとき, \n", "\n", "$$\n", "\\frac{n!}{a^{n^2}} = \\frac{n!}{(a^n)n} =\n", "\\frac{N!}{(a^n)^N}\\frac{N+1}{a^n}\\frac{N+2}{a^n}\\cdots\\frac{n}{a^n} \\leqq\n", "\\frac{N!}{(a^N)^n}\\left(\\frac{1}{2}\\right)^{n-N}\n", "\\to 0 \\quad (n\\to\\infty).\n", "\\quad \\QED\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### まとめ\n", "\n", "$n\\to\\infty$ のとき $\\ds\\frac{a_n}{b_n}\\to 0$ となることを $a_n\\prec b_n$ と書くことにする. $a>1$ のとき\n", "\n", "$$\n", "1 \\prec \\cdots \\prec \\log\\log n \\prec \\log n \\prec \n", "n \\prec n^2 \\prec\\cdots\\prec \n", "a^n \\prec n! \\prec a^{n^2}\\prec\\cdots\n", "$$\n", "\n", "右に行くほど $n\\to\\infty$ で速く増加する. このように増大度に階層性があるという認識は解析学における基本になる." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**例:** Stirlingの近似公式より, \n", "\n", "$$\n", "n! = n^n \\frac{1}{e^n} \\sqrt{n}\\;\\sqrt{2\\pi}\\;(1+\\eps_n) \\quad (\\eps_n\\to 0)\n", "$$\n", "\n", "が成立することが知られている(この一連のノート群の中でも複数回証明される)). 右辺に登場する $n^n$, $e^n$, $\\sqrt{n}$, $\\sqrt{2\\pi}$ は左のものほど $n\\to\\infty$ で速く増加する.\n", "\n", "$\\delta_n=\\log(1+\\eps_n)$ とおき, Stirlingの近似公式の両辺の対数を取ると, \n", "\n", "$$\n", "\\log n! = n\\log n - n + \\frac{1}{2}\\log n + \\frac{1}{2}\\log 2\\pi + \\delta_n\n", "\\quad (\\delta_n\\to 0).\n", "$$\n", "\n", "右辺に登場する $n\\log n$, $n$, $\\ds\\frac{1}{2}\\log n$, $\\ds\\frac{1}{2}\\log\\sqrt{2\\pi}$, $\\delta_n$ は左のものほど $n\\to\\infty$ で速く増加する. $\\QED$\n", "\n", "次のセルで $\\log n!$ がどのように近似されるかをプロットしてみよう. $\\QED$" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "plot_logfactorial (generic function with 1 method)" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "lfact(n) = lgamma(n+1) # = log n!\n", "f1(n) = n*log(n)\n", "f2(n) = n*log(n) - n\n", "f3(n) = n*log(n) - n + 0.5*log(n)\n", "f4(n) = n*log(n) - n + 0.5*log(n) + 0.5*log(2π)\n", "\n", "function plot_logfactorial(n)\n", " plot(legend=:topleft)\n", " plot!(n, lfact.(n), label=\"log n!\", lw=1.5)\n", " plot!(n, f1.(n), label=\"n log n\", ls=:dash)\n", " plot!(n, f2.(n), label=\"n log n - n\", ls=:dash)\n", " plot!(n, f3.(n), label=\"n log n - n + 0.5 log n\", ls=:dash)\n", " plot!(n, f4.(n), label=\"n log n - n + (1/2)log n + (1/2)log 2pi\", color=:red, ls=:dash)\n", "end" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot_logfactorial(10^16:10^16:10^18)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "すぐ上のプロットを見れば, $n$ がこのように極めて大きな値であれば $\\log n!$ の $\\log n^n = n\\log n$ による近似もそう悪くないことがわかる." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 上限と下限と上極限と下極限" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 上限と下限\n", "\n", "$Y$ は空でない実数の集合であると仮定する.\n", "\n", "実数 $M$ で任意の $y\\in Y$ について $y\\leqq M$ を満たすものを実数の集合 $Y$ の**上界**(upper bound)と呼ぶ. 実数 $m$ で任意の $y\\in Y$ について $m\\leqq y$ を満たすものを $Y$ の**下界**(lower bound)と呼ぶ.\n", "\n", "$Y$ の上界が1つ以上存在するとき, $Y$ は**上に有界**(bounded from above)であると言い, $Y$ の下界が1つ以上存在するとき, $Y$ は**下に有界**(bounded from below)であると言う. $Y$ が上に有界かつ下に有界であるとき, $Y$ は**有界**(bounded)であると言う.\n", "\n", "次の結果は認めて使うことにする.\n", "\n", "**実数の連続性:** 実数の空でない集合 $Y$ が上に有界ならば $Y$ の上界全体の集合の中に最小値が存在する. $\\QED$\n", "\n", "$Y$ が上に有界なとき, $Y$ の上界全体の集合の最小値を $Y$ の**最小上界**(least upper bound)もしくは**上限**(supremum)と呼び. 同様に, $Y$ が下に有界ならば $Y$ の下界全体の集合の中に最大値が存在する. その最大値を $Y$ の**最大下界**(greatest lower bound)または**下限**(infimum)と呼ぶ. $Y$ の上限と下限をそれぞれ\n", "\n", "$$\n", "\\sup Y, \\quad \\inf Y\n", "$$\n", "\n", "と書く. 集合 $X$ 上の実数値函数 $f(x)$ による集合 $X$ の像 $f(X)=\\{\\,f(x)\\mid x\\in X\\,\\}$ の上限と下限をそれぞれ\n", "\n", "$$\n", "\\sup f(X) = \\sup_{x\\in X}f(x), \\quad \\inf f(X) = \\inf_{x\\in X}f(x)\n", "$$\n", "\n", "と書く. $f(X)$ が上に(もしくは下)に有界なとき, 函数 $f$ は $X$ において上に有界(もしくは下)に有界であるという.\n", "\n", "$Y$ が上に有界でないときには $Y$ は(有限な)上限を持たないが, そのとき $\\sup Y = \\infty$ と書くことがある. 同様に $Y$ が下に有界でないときには $Y$ は(有限な)下限を持たないが, そのとき $\\inf Y=-\\infty$ と書くことがある.\n", "\n", "$Y$ に最大値 $\\max Y$ が存在するとき, $\\sup Y = \\max Y$ となる. 同様に $Y$ に最小値 $\\min Y$ が存在するとき, $\\inf Y = \\min Y$ となる.\n", "\n", "**上限と下限の便利なところ:** $X$ に最大値や最小値が存在しなくても, 上限と下限は値として $\\pm\\infty$ を許せば常に存在する. 上限と下限は, 最大値や最小値が存在しないときに, それらに代わるものとして用いることができる場合が結構ある. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**例:** 開区間 $Y = (0,1)$ には最小値も最大値も存在しない. $Y=(0,1)$ のとき $\\inf Y = 0$, $\\sup Y = 1$ となる. $\\QED$\n", "\n", "**例:** $Y=\\{1,1/2,1/3,\\ldots\\}$ には最大値 $1$ は存在するが, 最小値は存在しない. このとき, \n", "\n", "$$\n", "\\sup Y = \\sup_{n=1,2,3,\\ldots}\\frac{1}{n} = \\max_{n=1,2,3,\\ldots}\\frac{1}{n} = 1, \\quad\n", "\\inf Y = \\inf_{n=1,2,3,\\ldots}\\frac{1}{n} = 0.\n", "\\qquad \\QED\n", "$$\n", "\n", "**例:** $Y = \\{1,2,3,\\ldots\\}$ は上に有界でないので有限の上限を持たない. 下限は最小値の $1$ になる:\n", "\n", "$$\n", "\\sup\\{1,2,3,\\ldots\\} = \\infty, \\quad\n", "\\inf\\{1,2,3,\\ldots\\} = \\min\\{1,2,3,\\ldots\\} = 1.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 上極限と下極限\n", "\n", "この節はすぐに読む必要はない. 上極限と下極限が必要になったときに読めば十分である.\n", "\n", "実数の集合 $X$ とその部分集合 $Y$ を考える. $X$ の上界は $Y$ の上界にもなるので, $X$ の上界の最小値は $Y$ の上界の最小値以上になる. すなわち, $\\sup X\\geqq \\sup Y$ となる(これは図を描いて考えれば明らかだろう). 部分集合の上限はもとの集合の上限以下になる. 同様に部分集合の下限はもとの集合の下限以上になる. \n", "\n", "したがって, 実数列 $a_n$ に対して, 数列 $\\ds\\alpha_n = \\sup_{k\\geqq n} a_k$ は単調減少し, 数列 $\\ds\\beta_n = \\inf_{k\\geqq n} a_k$ は単調増加する. ゆえに $\\pm\\infty$ も収束先に含めれば, それらの数列は常に収束することになる. それぞれの収束先を次のように書く:\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\limsup_{n\\to\\infty} a_n = \\lim_{n\\to\\infty}\\sup_{k\\geqq n} a_k = \\inf_{n\\geqq 1}\\sup_{k\\geqq n} a_k,\n", "\\\\ &\n", "\\liminf_{n\\to\\infty} a_n = \\lim_{n\\to\\infty}\\inf_{k\\geqq n} a_k = \\sup_{n\\geqq 1}\\inf_{k\\geqq n} a_k.\n", "\\end{aligned}\n", "$$\n", "\n", "これらをそれぞれ**上極限**(limit superior), **下極限**(limit inferior)と呼ぶ. \n", "\n", "それぞれを $\\liminf a_n$, $\\limsup a_n$ と略して書くこともある.\n", "\n", "自明に $\\ds\\inf_{k\\geqq n} a_k\\leqq \\sup_{k\\geqq n} a_k$ が成立しているので, \n", "\n", "$$\n", "\\liminf_{n\\to\\infty} a_n \\leqq \\limsup_{n\\to\\infty} a_n\n", "$$\n", "\n", "が成立している." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**定理:** 実数列 $a_n$ について\n", "\n", "$$\n", "-\\infty < \\liminf_{n\\to\\infty} a_n = \\limsup_{n\\to\\infty} a_n < \\infty\n", "$$\n", "\n", "が成立しているならば, 実数列 $a_n$ は収束して,\n", "\n", "$$\n", "\\lim_{n\\to\\infty}a_n = \\liminf_{n\\to\\infty} a_n = \\limsup_{n\\to\\infty} a_n.\n", "$$\n", "\n", "**証明:** $\\alpha = \\liminf_{n\\to\\infty} a_n = \\limsup_{n\\to\\infty} a_n$ とおき, 任意に $\\eps>0$ と取って固定する. $\\alpha = \\liminf_{n\\to\\infty} a_n$ より, $\\inf_{j\\geqq n} a_j$ が $n$ について単調増加することに注意すれば, ある番号 $N'$ が存在して, \n", "\n", "$$\n", "\\alpha - \\eps \\leqq \\inf_{j\\geqq n} a_j \\leqq \\alpha \\quad (n\\geqq N').\n", "$$\n", "\n", "同様に, $\\sup_{j\\geqq n} a_j$ が $n$ について単調減少することに注意すれば, ある番号 $N''$ が存在して, \n", "\n", "$$\n", "\\alpha \\leqq \\sup_{j\\geqq n} a_j \\leqq \\alpha+\\eps \\quad (n\\geqq N'').\n", "$$\n", "\n", "ゆえに $n=\\max\\{N',N''\\}$ のとき, $k\\geqq n$ ならば\n", "\n", "$$\n", "\\alpha - \\eps \\leqq \\inf_{j\\geqq n} a_k \\leqq a_k\\leqq \\sup_{j\\geqq n} a_j \\leqq \\alpha+\\eps.\n", "$$\n", "\n", "これは $k\\geqq n$ のとき, $|a_k - \\alpha|\\leqq\\eps$ が成立することを意味する. したがって, 数列の収束の定義より, 数列 $a_n$ は $\\alpha$ に収束する. これで示すべきことが示された. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**上極限と下極限の便利な点:** $\\lim a_n$ と違って, $\\liminf a_n$ と $\\limsup a_n$ は収束先として $\\pm\\infty$ を含めれば常に収束しているので収束性を確認せずに気軽に使うことができる. そして上の定理より, $\\liminf a_n$ と $\\limsup a_n$ が一致していれば, $a_n$ は収束していて収束先はそれらと同じになる. \n", "\n", "だから, 以下のような使われ方をすることが多い. 実数列 $a_n$ に対して, 任意の $\\eps>0$ に対して, それに依存して決まる別の数列 $A_n^{(\\eps)}$ と $B_n^{(\\eps)}$ で同じ値 $\\alpha$ に収束するものが存在して, さらに\n", "\n", "$$\n", "A_n^{(\\eps)} - \\eps \\leqq a_n \\leqq B_n^{(\\eps)} + \\eps\n", "$$\n", "\n", "が成立していると仮定する. このとき, $n\\to\\infty$ とすると,\n", "\n", "$$\n", "\\alpha-\\eps \\leqq\\liminf_{n\\to\\infty}a_n\\leqq\\limsup_{n\\to\\infty}a_n\\leqq \\alpha+\\eps.\n", "$$\n", "\n", "$\\eps>0$ は幾らでも小さくできるので,\n", "\n", "$$\n", "\\liminf_{n\\to\\infty}a_n = \\limsup_{n\\to\\infty}a_n = \\alpha\n", "$$\n", "\n", "となる. ゆえに数列 $a_n$ は $\\alpha$ に収束する:\n", "\n", "$$\n", "\\lim_{n\\to\\infty} a_n = \\alpha.\n", "$$\n", "\n", "「$n\\to\\infty$ とすると」の段階では数列 $a_n$ が収束するかどうかわかっていないので, $\\lim$ ではなく, $\\liminf$ と $\\limsup$ を使わなければいけない. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 以下の実数列の上極限を求めよ.\n", "\n", "(1) $a_n = 10/n + (-1)^n$.\n", "\n", "(2) $b_n = 10/n + \\sin(4n)$. \n", "\n", "$\\pi$ が無理数であることを認めて使ってよい.\n", "\n", "**解答例:** (1) $\\limsup a_n = \\limsup(10/n + (-1)^n) = 1$.\n", "\n", "(2) $\\pi$ は無理数なので $a = 4/(2\\pi)$ も無理数である. そのとき, $\\sin(4n)=\\sin(2\\pi na)$. 次の問題より, 任意の $k>0$ に対して, ある正の整数 $n_k$ で $n_k a$ の小数点以下の部分と $1/4$ の距離が $1/k$ 以下になるものが存在する. そのとき, $n_ka$ の小数点以下の部分は $1/4$ に収束するので, $k\\to\\infty$ で $\\sin(2\\pi n_k a)\\to \\sin(2\\pi(1/4))=1$ となる. これより, $\\limsup b_n = 1$ となることがわかる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 実数 $x$ に対して $x$ 以下の最大の整数を $\\lfloor x\\rfloor$ と書き, $f(x)=x-\\lfloor x\\rfloor$ とおく. $f(x)$ は $x$ の小数点以下の部分になっている. $a$ は無理数であるとする. そのとき, 任意の $\\alpha\\in[0,1)$ と $\\eps>0$ に対して, ある正の整数 $n$ で $f(na)$ と $\\alpha|$ の距離が $\\eps$ 以下になるものが存在することを示せ.\n", "\n", "**解答例:** $a$ は無理数なので $a,2a,3a,\\ldots$ は決して整数にはならない. そして, $f(a), f(2a), f(3a),\\ldots$ の中の任意の2つは決して一致しない. なぜならば, もしも $f(ma)=f(na)$, $m\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nmax = 100\n", "n = 1:nmax\n", "plot(n, @.(10/n + (-1)^n), label=\"a_n\", ylims=(-1.5,10), marker=:o, markersize=3, markerstrokealpha=0)\n", "plot!(n, fill(1, size(n)), label=\"1\")" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "image/png": 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"\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nmax = 200\n", "n = 1:nmax\n", "plot(n, @.(10/n + sin(4*n)), label=\"b_n\", ylims=(-1.5, 10), marker=:o, markersize=2, markerstrokealpha=0)\n", "plot!(n, fill(1, size(n)), label=\"1\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題(上極限の特徴付け):** $a_n$ は実数列であるとし, $\\ds\\alpha = \\limsup_{n\\to\\infty}a_n$ とおく. 以下を示せ.\n", "\n", "(1) $\\alpha > A$ のとき, 任意の番号 $n$ に対して, ある $k\\geqq n$ で $a_k \\geqq A$ を満たすものが存在する($a_k\\geqq A$ を満たす $k$ が無限個存在する).\n", "\n", "(2) $\\alpha < B$ のとき, ある番号 $N$ が存在して, $k\\geqq N$ ならば $a_k\\leqq B$ となる(ある番号から先のすべての $k$ について $a_k\\leqq B$ が成立している).\n", "\n", "逆に(1),(2)を $\\alpha$ が満たしていれば, $\\ds\\alpha = \\limsup_{n\\to\\infty}a_n$ となることも示せ.\n", "\n", "**解答例:** (1)を示そう. $\\ds \\alpha=\\inf_{n\\geqq 1}\\sup_{k\\geqq n} a_k$ より, 任意の番号 $n$ について $\\ds \\sup_{k\\geqq n}a_k\\geqq\\alpha$ となる. ゆえに, $\\alpha > A$ という仮定より, 各 $n$ ごとにある $k\\geqq n$ で $a_k\\geqq A$ を満たすものが存在する. \n", "\n", "(2)を示そう. $\\alpha < B$ と $\\ds\\sup_{k\\geqq n}\\to \\alpha$ より, ある番号 $N$ が存在して, $n\\geqq N$ ならば $\\ds\\sup_{k\\geqq n}a_k\\leqq B$ すなわち $a_k\\leqq B$ ($k\\geqq n$) となる. ゆえに $k\\geqq N$ ならば $a_k\\leqq B$ となる.\n", "\n", "(1),(2)を仮定して, $\\ds\\alpha = \\limsup_{n\\to\\infty}a_n$ を示そう. $A < \\alpha < B$ と仮定する. (1)より, 任意の番号 $n$ に対してある $k\\geqq n$ で $a_k\\geqq A$ を満たすものが存在するので, $\\ds\\sup_{k\\geqq n}a_k\\geqq A$ となる. ゆえに $\\ds\\limsup_{n\\to\\infty}a_n\\geqq A$ となる. (2)より, ある番号 $N$ が存在して, $k\\geqq N$ ならば $a_k\\leqq B$ となる. すなわち, $\\ds\\sup_{k\\geqq N}\\leqq B$ となる. \n", "ゆえに, $\\ds\\limsup_{n\\to\\infty}a_n\\leqq B$ となる. これで\n", "\n", "$$\n", "A\\leqq \\limsup_{n\\to\\infty} a_n\\leqq B\n", "$$\n", "\n", "が示された. $A,B$ は $\\alpha$ に幾らででも近くに取れるので $\\ds\\alpha = \\limsup_{n\\to\\infty}a_n$ となる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 実数列 $a_n$ が収束しているならば\n", "\n", "$$\n", "\\limsup_{n\\to\\infty} a_n = \\liminf_{n\\to\\infty} a_n = \\lim_{n\\to\\infty}a_n\n", "$$\n", "\n", "となっていることを示せ.\n", "\n", "**解答例:** 実数列 $a_n$ は $\\alpha$ に収束していると仮定する. $\\ds\\limsup_{n\\to\\infty} a_n=\\alpha$ のみを示そう. $A<\\alpha0$ を取る. もしもすべての番号 $n$ について $a_n\\leqq \\alpha-\\eps$ となっているならば,$\\alpha-\\eps$ も $a_n$ 達の上界になり, $\\alpha$ が最小の上界であることに反するので, ある番号 $N$ が存在して $a_N > \\alpha-\\eps$ となる. $a_n$ は単調増加数列なので $n\\geqq N$ のとき $a_n>\\alpha-\\eps$ となる. $\\alpha$ は $a_n$ たちの上界なので $a_n\\leqq \\alpha$ でもある. ゆえに $n\\geqq N$ ならば, $\\alpha-\\eps0$ に対して, ある番号 $N$ が存在して, $m,n\\geqq N$ ならば $|a_m-a_n|<\\eps$ が存在することであると定める.\n", "\n", "**実数のCauchy列の収束:** 実数のCauchy列はある実数に収束する.\n", "\n", "**証明:** $a_n$ は実数のCauchy列であるとする. そのとき, ある番号 $N$ が存在して $n\\geqq N$ のとき $|a_n-a_N|< 1$ となるので, $|a_n|< |a_N|+1$ となる. ゆえに $M=\\max\\{|a_1|,\\ldots,|a_{N-1}|, |a_N|+1\\}$ とおくと, すべての番号 $n$ について $|a_n|\\leqq M$ となり, 数列 $a_n$ が有界であることがわかる. \n", "\n", "Bolzano-Weierstrass の定理より, $a_n$ のある部分列 $a_{k_n}$ である実数 $\\alpha$ に収束するものが存在する. $\\eps>0$ を任意に取る. $a_n$ はCauchy列なので, ある番号 $N'$ が存在して $m,n\\geqq N'$ ならば $\\ds|a_n-a_m|<\\frac{\\eps}{2}$ となる. 部分列 $a_{k_n}$ は $\\alpha$ に収束するので, ある番号 $N''$ が存在して $m\\geqq N''$ ならば $\\ds|a_{k_m}-\\alpha|<\\frac{\\eps}{2}$ となる. $k_m\\geqq N'$ となるような $m\\geqq N''$ が存在する. そのとき, $n\\geqq N'$ ならば\n", "\n", "$$\n", "|a_n-\\alpha|\\leqq|a_n-a_{k_m}|+|a_{k_m}-\\alpha|<\\frac{\\eps}{2}+\\frac{\\eps}{2}=\\eps.\n", "$$\n", "\n", "これで, $a_n$ が $\\alpha$ に収束することが示された. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 閉区間に関するHeine-Borelの被覆定理\n", "\n", "**閉区間に関するHeine-Borelの被覆定理:** $a\\leqq b$ であるとし, 閉区間 $I=[a,b]$ を考える. 開区間 $U_\\lambda=(a_\\lambda,,b_\\lambda)$, $a_\\lambda連続函数」の対応する節を参照せよ." ] } ], "metadata": { "@webio": { "lastCommId": null, "lastKernelId": null }, "_draft": { "nbviewer_url": "https://gist.github.com/746254afb64b8be4e0cf6990cf321a1f" }, "gist": { "data": { "description": "01 convergence", "public": true }, "id": "746254afb64b8be4e0cf6990cf321a1f" }, "jupytext": { "formats": "ipynb,md" }, "kernelspec": { "display_name": "Julia 1.9.1", "language": "julia", "name": "julia-1.9" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.9.1" }, "toc": { "base_numbering": 1, "nav_menu": { "height": "207px", "width": "242px" }, "number_sections": true, "sideBar": true, "skip_h1_title": true, "title_cell": "目次", "title_sidebar": "目次", "toc_cell": true, "toc_position": { "height": "calc(100% - 180px)", "left": "10px", "top": "150px", "width": "214px" }, "toc_section_display": true, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 2 }