{"id":6673,"date":"2021-07-20T23:49:14","date_gmt":"2021-07-20T14:49:14","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6673"},"modified":"2021-07-22T21:12:08","modified_gmt":"2021-07-22T12:12:08","slug":"geometric-sequences","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/geometric-sequences\/","title":{"rendered":"\ub4f1\ube44\uc218\uc5f4\uc758 \uadf9\ud55c"},"content":{"rendered":"
\n

\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 1\uc7a5 4\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.  (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n

\ub4f1\ube44\uc218\uc5f4\uc758 \uadf9\ud55c\uc740 \ubb34\ud55c\uae09\uc218\uc640 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc131\uc9c8\uc744 \ubc1d\ud790 \ub54c \uc911\uc694\ud55c \uc5ed\ud560\uc744 \ud55c\ub2e4.<\/p>\n

\n

\uc815\ub9ac 1.4.1.<\/span>
\n\\(r\\)\uac00 \uc2e4\uc218\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

    \n
  1. \ub9cc\uc57d \\(\\lvert r \\rvert < 1\\)\uc774\uba74 \uc218\uc5f4 \\(\\left\\{ r^n \\right\\}\\)\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/li>\n
  2. \ub9cc\uc57d \\(r= 1\\)\uc774\uba74 \uc218\uc5f4 \\(\\left\\{ r^n \\right\\}\\)\uc774 \\(1\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/li>\n
  3. \ub9cc\uc57d \\(r \\le -1\\)\uc774\uba74 \uc218\uc5f4 \\(\\left\\{ r^n \\right\\}\\)\uc774 \uc9c4\ub3d9\ud55c\ub2e4.<\/li>\n
  4. \ub9cc\uc57d \\(r > 1\\)\uc774\uba74 \uc218\uc5f4 \\(\\left\\{ r^n \\right\\}\\)\uc774 \uc591\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
    \n

    \uc99d\uba85.<\/p>\n

    [1] \\(0 < r < 1\\)\uc778 \uacbd\uc6b0\ub97c \uba3c\uc800 \uc0b4\ud3b4\ubcf4\uc790.\n\\[h = \\frac{1}{r} -1\\]\n\uc774\ub77c\uace0 \ud558\uba74 \\(h > 0\\)\uc774\uace0
    \n\\[r = \\frac{1}{1+h}\\]
    \n\uc774\ub2e4. \ubca0\ub974\ub204\uc774 \ubd80\ub4f1\uc2dd(Bernoulli’s inequality)\uc5d0 \uc758\ud558\uc5ec, \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[(1+h)^n \\ge 1+nh\\]
    \n\uc774\ubbc0\ub85c
    \n\\[0\\le r^n = \\left( \\frac{1}{1+h} \\right)^n = \\frac{1}{(1+h)^n} \\le \\frac{1}{1+nh}\\]
    \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7f0\ub370 \ub9c8\uc9c0\ub9c9 \ubd84\uc218\uc2dd\uc740 \\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc870\uc784 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(r^n \\,\\rightarrow\\,0\\)\uc774\ub2e4.<\/p>\n

    \ub9cc\uc57d \\(r=0\\)\uc774\uba74 \\(n\\rightarrow\\infty\\)\uc77c \ub54c \ub2f9\uc5f0\ud788 \\(r^n \\,\\rightarrow\\,0\\)\uc774\ub2e4.<\/p>\n

    \\(-1 < r < 0\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(0 < \\lvert r \\rvert < 1\\)\uc774\ubbc0\ub85c\n\\[ - \\lvert r \\rvert^n \\le r^n \\le \\lvert r \\rvert^n\\]\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(n\\rightarrow\\infty\\)\uc77c \ub54c\n\\[ -\\lvert r \\rvert ^n \\,\\rightarrow\\, 0 \\quad \\text{and}\\quad \\lvert r \\rvert \\,\\rightarrow\\, 0\\]\n\uc774\ubbc0\ub85c, \uc870\uc784 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(r^n \\,\\rightarrow\\,0\\)\uc774\ub2e4.<\/p>\n

    [2] \\(r=1\\)\uc774\uba74 \\(n\\rightarrow\\infty\\)\uc77c \ub54c \uc790\uba85\ud558\uac8c \\(r^n = 1 \\,\\rightarrow\\, 1\\)\uc774\ub2e4.<\/p>\n

    [3] \\(r\\le -1\\)\uc774\uba74, \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[r^{2n} \\ge 1 \\quad\\text{and}\\quad r^{2n+1} \\le -1\\]
    \n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{r^n\\right\\}\\)\uc740 \ud558\ub098\uc758 \uac12\uc5d0 \uac00\uac00\uc6cc\uc9c8 \uc218 \uc5c6\uace0, \uc591\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud558\uac70\ub098 \uc74c\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud558\uc9c0\ub3c4 \uc54a\ub294\ub2e4.<\/p>\n

    [4] \\(r > 1\\)\uc778 \uacbd\uc6b0 \\(h = r-1\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(h > 0\\)\uc774\uace0 \\(r = 1+h\\)\uc774\ub2e4. \ubca0\ub974\ub204\uc774 \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[r^n = (1+h)^n \\ge 1+nh\\]
    \n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(1+nh \\,\\rightarrow\\,\\infty\\)\uc774\ubbc0\ub85c, \uadf9\ud55c\uc758 \uc21c\uc11c \ubcf4\uc874 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec, \\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(r^n \\,\\rightarrow\\,\\infty\\)\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \n

    \uc608\uc81c 1.4.1.<\/span>
    \n\ub2e4\uc74c \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uc870\uc0ac\ud558\uc2dc\uc624.
    \n\\[\\left\\{ \\frac{2^{n+1}}{3^n +4} \\right\\}\\]\n<\/p>\n

    \ud480\uc774.<\/span>
    \n\\[\\lim_{n\\rightarrow\\infty} \\frac{2^{n+1}}{3^n +4}
    \n= \\lim_{n\\rightarrow\\infty} \\frac{\\left(\\frac{2}{3}\\right)^n \\times 2}{1+\\frac{4}{3^n}} = \\frac{0\\times 2}{1+0} = 0.
    \n\\]\n<\/p>\n<\/div>\n

    \n

    \uc608\uc81c 1.4.2.<\/span>
    \n\ub2e4\uc74c \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uc870\uc0ac\ud558\uc2dc\uc624.
    \n\\[\\left\\{ \\frac{4^n – 2^n}{3^n + 2^n} \\right\\}\\]\n<\/p>\n

    \ud480\uc774.<\/span>
    \n\uc8fc\uc5b4\uc9c4 \uc218\uc5f4\uc758 \uc77c\ubc18\ud56d\uc744 \ubcc0\ud615\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
    \n\\[\\frac{4^n – 2^n}{3^n +2^n} = \\left(\\frac{4}{3}\\right)^n \\times \\frac{1 – \\left(\\frac{2}{4}\\right)^n}{1+ \\left(\\frac{2}{3}\\right)^n} .\\]
    \n\uadf8\ub7f0\ub370 \\(n\\rightarrow\\infty\\)\uc77c \ub54c
    \n\\[\\frac{1 – \\left(\\frac{2}{4}\\right)^n}{1+ \\left(\\frac{2}{3}\\right)^n} \\,\\rightarrow\\,1\\]
    \n\uc774\ubbc0\ub85c, \uc720\ud55c \uac1c\ub97c \uc81c\uc678\ud55c \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[\\frac{1 – \\left(\\frac{2}{4}\\right)^n}{1+ \\left(\\frac{2}{3}\\right)^n} \\ge \\frac{1}{2}\\]
    \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc989 \uc720\ud55c \uac1c\ub97c \uc81c\uc678\ud55c \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[\\left(\\frac{4}{3}\\right)^n \\cdot \\frac{1 – \\left(\\frac{2}{4}\\right)^n}{1+ \\left(\\frac{2}{3}\\right)^n} \\ge \\left(\\frac{4}{3}\\right)^n \\cdot \\frac{1}{2}\\]
    \n\uc774\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc740 \\(n\\rightarrow\\infty\\)\uc77c \ub54c \uc591\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uadf9\ud55c\uc758 \uc21c\uc11c \ubcf4\uc874 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \uc774 \ubd80\ub4f1\uc2dd\uc758 \uc88c\ubcc0\ub3c4 \\(n\\rightarrow\\infty\\)\uc77c \ub54c \uc591\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud55c\ub2e4. \uc989
    \n\\[ \\lim_{n\\rightarrow\\infty} \\frac{4^n – 2^n}{3^n + 2^n} = \\infty \\]
    \n\uc774\ub2e4.\n<\/p>\n<\/div>\n

    <\/p>\n

    \n\\[
    \n\\newcommand{\\Hom}{{\\operatorname{Hom}}}
    \n\\newcommand{\\Mat}{{\\operatorname{Mat}}}
    \n\\newcommand{\\proj}{{\\operatorname{proj}}}
    \n\\newcommand{\\adj}{{\\operatorname{adj}}}
    \n\\]\n<\/div>\n
    \n