プロローグ
どうも、安田です。今回は数列$\{Y_n\}$の続きです。
この記事はシリーズ記事です。是非p1からご覧ください。
このシリーズの最初の記事
このシリーズの前回記事
問題:数列$\{Y_n\}$
このシリーズでの最終目的は、以下の数列の一般項を求める事です。性質などの発見もサブの目的として取り扱っていきます。
考察
積分近似
どうも、安田です。
前回前々回となかなか一般項を求める事が出来なかったため、今回は近似からアプローチします。
数列$Y$を考える際に、漸化式において前項から次項へと$\frac{1}{Y_n}$だけ進むという事に着目します。
$y=\frac{1}{x}$のグラフを考えた時、前項$x$から次項$x'$へ進むためには, $\frac{1}{x}$だけ進む必要があります。これをグラフで見ると以下のようになります。
という感じです。
ここで、上のグラフにおいて面積を考えます。前項を$Y_k$として、$(Y_k, 0)$, $(Y_k, \frac{1}{Y_k})$, $(Y_{k+1}, 0)$, $(Y_{k+1}, \frac{1}{Y_{k+1}})$で囲まれた台形と、$x$軸, $y=\frac{1}{x}$, $x=Y_k$, $x=Y_{k+1}$で囲まれた面積がおおよそ一致する事から近似式を立てます。
より漸化式が複雑になりました。
実を言うと、積分近似は他の数列に応用すると凄い事が起こるのですが、これは別の記事にする予定ですので、ここでは書きません。
微分近似
次は積分と対比的に微分でアプローチします。
ここでは$Y$の漸化式の形に着目します。
これは数列$Y$の増加量を表していますので、これを傾きと見立てて微分方程式の近似式を立てる事ができます。
漸化式より \begin{eqnarray*} Y_{x+1} - Y_x &=& \frac{1}{Y_x} \\ g((x+1)h) - g(xh) &=& \frac{1}{g(xh)} \\ \frac{g(xh+h) - g(xh)}{h} &=& \frac{1}{hg(xh)} \end{eqnarray*} $xh=t$とおくと \begin{eqnarray*} \frac{g(t+h) - g(t)}{h} &=& \frac{1}{hg(t)} \\ \frac{d}{dt} g(t) &\simeq& \frac{1}{hg(t)} \\ g(t) dg(t) &\simeq& \frac{dt}{h} \\ \frac{1}{2}g(t)^2 &\simeq& \frac{t}{h} + C \\ g(t) &\simeq& \sqrt{2\frac{t}{h} + 2C} \end{eqnarray*} $t = xh$を代入して \begin{equation} g(xh) \simeq \sqrt{2x + 2C} \end{equation} すなわち, \begin{equation} Y_n \simeq \sqrt{2n + 2C} \end{equation} である。ここで、$Y_1 = 2$の初期条件で一致させると$C=1$である。ゆえに \begin{equation} Y_n \simeq \sqrt{2n + 2} \end{equation} である。
実際に計算できる近似式が求まりましたね。
この時、近似式と$Y_n$のグラフは以下のようになります。
誤差はまあまあ小さめで、$n=1$のとき一致し、$n=6$の時は$0.1$に満たないほどです。
また、これを積分近似で求めた近似式に代入してみます。

となった。
非常に誤差が小さい事が分かります。もしかしたら、微分近似と積分近似の相性はお互いの悪い部分を相殺し合うのかもしれませんね。
皆さんも、グラフから数列の近似式を考えるという事を一回試してみると面白いかもしれません。
それでは、お疲れさまでした。
数列Yシリーズ

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