{"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":"2026-09-27T17:12:07","modified_gmt":"2026-09-27T08:12:07","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

\uc2e4\uc218\uc5f4 \\(\\left\\{a_n\\right\\}\\)\uc774 \uc5b4\ub5a4 \uc2e4\uc218 \\(a,\\) \\(r\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[
\na_n=ar^{\\,n-n_0}
\n\\]
\n\uc758 \uaf34\ub85c \ub098\ud0c0\ub098\uba74 \\(\\left\\{a_n\\right\\}\\)\uc744 \ub4f1\ube44\uc218\uc5f4<\/span>(geometric sequence)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc5ec\uae30\uc11c \\(a=a_{n_0}\\)\ub97c \ucd08\ud56d, \\(r\\)\ub97c \uacf5\ube44<\/span>(common ratio)\ub77c\uace0 \ubd80\ub978\ub2e4. \ub4f1\ube44\uc218\uc5f4\uc758 \uadf9\ud55c\uc740 \uacf5\ube44 \\(r\\)\uc5d0 \ub530\ub978 \uc218\uc5f4 \\(\\left\\{r^n\\right\\}\\)\uc758 \uadf9\ud55c\uc744 \uc54c\uba74 \uc870\uc0ac\ud560 \uc218 \uc788\ub2e4. \uc774 \uacb0\uacfc\ub294 \ub4a4\uc5d0\uc11c \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

\ubcf4\uc870\uc815\ub9ac. (\ubca0\ub974\ub204\uc774 \ubd80\ub4f1\uc2dd)<\/span><\/p>\n

\\(h>0\\)\uc774\uace0 \\(n\\)\uc774 \uc591\uc758 \uc815\uc218\uc774\uba74
\n\\[
\n(1+h)^n\\ge1+nh
\n\\]
\n\uc774\ub2e4.<\/p>\n<\/div>\n

\n

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

\uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uc790. \\(n=1\\)\uc77c \ub54c\uc5d0\ub294
\n\\[
\n1+h=1+h
\n\\]
\n\uc774\ubbc0\ub85c \uc131\ub9bd\ud55c\ub2e4. \uc5b4\ub5a4 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[
\n(1+h)^n\\ge1+nh
\n\\]
\n\ub77c\uace0 \uac00\uc815\ud558\uc790. \\(1+h>0\\)\uc774\ubbc0\ub85c
\n\\[
\n\\begin{aligned}
\n(1+h)^{n+1}
\n&=(1+h)^n(1+h)\\\\
\n&\\ge(1+nh)(1+h)\\\\
\n&=1+(n+1)h+nh^2\\\\
\n&\\ge1+(n+1)h.
\n\\end{aligned}
\n\\]
\n\ub530\ub77c\uc11c \ubaa8\ub4e0 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \ubca0\ub974\ub204\uc774 \ubd80\ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n<\/span><\/p>\n<\/div>\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\\}\\)\uc740 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/li>\n
  2. \ub9cc\uc57d \\(r=1\\)\uc774\uba74 \uc218\uc5f4 \\(\\left\\{r^n\\right\\}\\)\uc740 \\(1\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/li>\n
  3. \ub9cc\uc57d \\(r\\le-1\\)\uc774\uba74 \uc218\uc5f4 \\(\\left\\{r^n\\right\\}\\)\uc740 \uc9c4\ub3d9\ud55c\ub2e4.<\/li>\n
  4. \ub9cc\uc57d \\(r>1\\)\uc774\uba74 \uc218\uc5f4 \\(\\left\\{r^n\\right\\}\\)\uc740 \uc591\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
    \n

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

    [1] \uba3c\uc800
    \n\\[
    \n0 < r < 1\n\\]\n\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uacfc \uac19\uc774 \ub193\ub294\ub2e4.\n\\[\nh=\\frac{1}{r}-1.\n\\]\n\uadf8\ub7ec\uba74 \\(h>0\\)\uc774\uace0
    \n\\[
    \nr=\\frac{1}{1+h}.
    \n\\]
    \n\ubca0\ub974\ub204\uc774 \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec
    \n\\[
    \n(1+h)^n\\ge1+nh
    \n\\]
    \n\uc774\ubbc0\ub85c
    \n\\[
    \n0\\le r^n
    \n=
    \n\\frac{1}{(1+h)^n}
    \n\\le
    \n\\frac{1}{1+nh}.
    \n\\]
    \n\uadf8\ub7f0\ub370
    \n\\[
    \n\\lim_{n\\rightarrow\\infty}
    \n\\frac{1}{1+nh}
    \n=
    \n\\lim_{n\\rightarrow\\infty}
    \n\\frac{\\frac{1}{n}}{\\frac{1}{n}+h}
    \n=
    \n0.
    \n\\]
    \n\ub530\ub77c\uc11c \uc870\uc784 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec
    \n\\[
    \n\\lim_{n\\rightarrow\\infty}r^n=0.
    \n\\]\n<\/p>\n

    \\(r=0\\)\uc774\uba74 \ubaa8\ub4e0 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(r^n=0\\)\uc774\ubbc0\ub85c \uc790\uba85\ud558\uac8c
    \n\\[
    \n\\lim_{n\\rightarrow\\infty}r^n=0.
    \n\\]\n<\/p>\n

    \ub9c8\uc9c0\ub9c9\uc73c\ub85c
    \n\\[
    \n-1 < r < 0\n\\]\n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74\n\\[\n0 < |r| < 1\n\\]\n\uc774\ubbc0\ub85c \uc55e\uc5d0\uc11c \uc99d\uba85\ud55c \uacb0\uacfc\uc5d0 \ub530\ub77c\n\\[\n|r|^n\\rightarrow0.\n\\]\n\ub610\ud55c \ubaa8\ub4e0 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\n-|r|^n\\le r^n\\le|r|^n\n\\]\n\uc774\uace0\n\\[\n-|r|^n\\rightarrow0,\n\\qquad\n|r|^n\\rightarrow0.\n\\]\n\ub530\ub77c\uc11c \uc870\uc784 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec\n\\[\nr^n\\rightarrow0.\n\\]\n<\/p>\n

    [2] \\(r=1\\)\uc774\uba74 \ubaa8\ub4e0 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \nr^n=1
    \n\\]
    \n\uc774\ubbc0\ub85c
    \n\\[
    \n\\lim_{n\\rightarrow\\infty}r^n=1.
    \n\\]\n<\/p>\n

    [3] \\(r\\le-1\\)\uc774\ub77c\uace0 \ud558\uc790. \ubaa8\ub4e0 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \nr^{2n}\\ge1
    \n\\]
    \n\uc774\uace0
    \n\\[
    \nr^{2n+1}\\le-1.
    \n\\]
    \n\ub530\ub77c\uc11c \uc774 \uc218\uc5f4\uc740 \uc5b4\ub5a0\ud55c \uc2e4\uc218\uc5d0\ub3c4 \uc218\ub834\ud560 \uc218 \uc5c6\ub2e4. \uc2e4\uc81c\ub85c \uc5b4\ub5a4 \uc2e4\uc218 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \\(L\\ge0\\)\uc774\uba74 \ubaa8\ub4e0 \ud640\uc218 \ubc88\uc9f8 \ud56d\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n|r^{2n+1}-L|\\ge1
    \n\\]
    \n\uc774\uace0, \\(L< 0\\)\uc774\uba74 \ubaa8\ub4e0 \uc9dd\uc218 \ubc88\uc9f8 \ud56d\uc5d0 \ub300\ud558\uc5ec\n\\[\n|r^{2n}-L|\\ge1.\n\\]\n\uc5b4\ub290 \uacbd\uc6b0\uc5d0\ub3c4 \uc218\ub834\uc758 \uc815\uc758\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<\/p>\n

    \ub610\ud55c \ud640\uc218 \ubc88\uc9f8 \ud56d\uc740 \ud56d\uc0c1 \\(-1\\) \uc774\ud558\uc774\ubbc0\ub85c \uc218\uc5f4 \uc804\uccb4\uac00 \uc591\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud560 \uc218 \uc5c6\uace0, \uc9dd\uc218 \ubc88\uc9f8 \ud56d\uc740 \ud56d\uc0c1 \\(1\\) \uc774\uc0c1\uc774\ubbc0\ub85c \uc74c\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud560 \uc218\ub3c4 \uc5c6\ub2e4. \ub530\ub77c\uc11c \uc774 \ucc45\uc758 \uc6a9\uc5b4\uc5d0 \ub530\ub77c \\(\\left\\{r^n\\right\\}\\)\uc740 \uc9c4\ub3d9\ud55c\ub2e4.<\/p>\n

    [4] \\(r>1\\)\uc774\ub77c\uace0 \ud558\uc790. \\(h=r-1\\)\ub85c \ub193\uc73c\uba74 \\(h>0\\)\uc774\uace0 \\(r=1+h\\)\uc774\ub2e4. \ubca0\ub974\ub204\uc774 \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec
    \n\\[
    \nr^n=(1+h)^n\\ge1+nh.
    \n\\]
    \n\uadf8\ub7f0\ub370
    \n\\[
    \n1+nh\\rightarrow\\infty
    \n\\]
    \n\uc774\ubbc0\ub85c \uadf9\ud55c\uc758 \uc21c\uc11c \ubcf4\uc874 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec
    \n\\[
    \nr^n\\rightarrow\\infty.
    \n\\]
    \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\\[
    \n\\left\\{
    \n\\frac{2^{n+1}}{3^n+4}
    \n\\right\\}.
    \n\\]\n<\/p>\n

    \ud480\uc774.<\/span>
    \n\ubd84\uc790\uc640 \ubd84\ubaa8\ub97c \\(3^n\\)\uc73c\ub85c \ub098\ub204\uba74
    \n\\[
    \n\\frac{2^{n+1}}{3^n+4}
    \n=
    \n\\frac{
    \n2\\left(\\frac{2}{3}\\right)^n
    \n}{
    \n1+4\\left(\\frac{1}{3}\\right)^n
    \n}.
    \n\\]
    \n\uc815\ub9ac 1.4.1\uc5d0 \uc758\ud558\uc5ec
    \n\\[
    \n\\left(\\frac{2}{3}\\right)^n\\rightarrow0,
    \n\\qquad
    \n\\left(\\frac{1}{3}\\right)^n\\rightarrow0.
    \n\\]
    \n\ub530\ub77c\uc11c
    \n\\[
    \n\\begin{aligned}
    \n\\lim_{n\\rightarrow\\infty}
    \n\\frac{2^{n+1}}{3^n+4}
    \n&=
    \n\\frac{2\\cdot0}{1+4\\cdot0}\\\\
    \n&=0.
    \n\\end{aligned}
    \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\\[
    \n\\left\\{
    \n\\frac{4^n-2^n}{3^n+2^n}
    \n\\right\\}.
    \n\\]\n<\/p>\n

    \ud480\uc774.<\/span>
    \n\uc8fc\uc5b4\uc9c4 \uc218\uc5f4\uc758 \uc77c\ubc18\ud56d\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.
    \n\\[
    \n\\frac{4^n-2^n}{3^n+2^n}
    \n=
    \n\\left(\\frac{4}{3}\\right)^n
    \n\\frac{
    \n1-\\left(\\frac{1}{2}\\right)^n
    \n}{
    \n1+\\left(\\frac{2}{3}\\right)^n
    \n}.
    \n\\]
    \n\uc815\ub9ac 1.4.1\uc5d0 \uc758\ud558\uc5ec
    \n\\[
    \n\\left(\\frac{1}{2}\\right)^n\\rightarrow0,
    \n\\qquad
    \n\\left(\\frac{2}{3}\\right)^n\\rightarrow0.
    \n\\]
    \n\ub530\ub77c\uc11c
    \n\\[
    \n\\frac{
    \n1-\\left(\\frac{1}{2}\\right)^n
    \n}{
    \n1+\\left(\\frac{2}{3}\\right)^n
    \n}
    \n\\rightarrow1.
    \n\\]
    \n\uadf8\ub7ec\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n\\frac{
    \n1-\\left(\\frac{1}{2}\\right)^n
    \n}{
    \n1+\\left(\\frac{2}{3}\\right)^n
    \n}
    \n\\ge\\frac12.
    \n\\]
    \n\ub530\ub77c\uc11c \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n\\frac{4^n-2^n}{3^n+2^n}
    \n\\ge
    \n\\frac12
    \n\\left(\\frac{4}{3}\\right)^n.
    \n\\]
    \n\ud55c\ud3b8 \uc815\ub9ac 1.4.1\uc5d0 \uc758\ud558\uc5ec
    \n\\[
    \n\\left(\\frac{4}{3}\\right)^n\\rightarrow\\infty
    \n\\]
    \n\uc774\ubbc0\ub85c
    \n\\[
    \n\\frac12
    \n\\left(\\frac{4}{3}\\right)^n
    \n\\rightarrow\\infty.
    \n\\]
    \n\ub530\ub77c\uc11c \uadf9\ud55c\uc758 \uc21c\uc11c \ubcf4\uc874 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec
    \n\\[
    \n\\lim_{n\\rightarrow\\infty}
    \n\\frac{4^n-2^n}{3^n+2^n}
    \n=
    \n\\infty.
    \n\\]\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