{"id":6669,"date":"2021-07-20T23:48:12","date_gmt":"2021-07-20T14:48:12","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6669"},"modified":"2021-08-21T18:28:46","modified_gmt":"2021-08-21T09:28:46","slug":"limit-of-a-sequence","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/limit-of-a-sequence\/","title":{"rendered":"\uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc815\uc758"},"content":{"rendered":"
\n

\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 1\uc7a5 2\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.  (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n

<\/p>\n

\uc218\ub834\ud558\ub294 \uc218\uc5f4<\/h2>\n

\\(\\left\\{ a_n \\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\uace0 \\(L\\)\uc774 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(n\\)\uc774 \ud55c\uc5c6\uc774 \ucee4\uc9c8 \ub54c \ud56d \\(a_n\\)\uc774 \\(L\\)\uc5d0 \ud55c\uc5c6\uc774 \uac00\uae4c\uc6cc\uc9c0\uba74 \u201c\\(\\left\\{ a_n \\right\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834<\/span>(converge)\ud55c\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc774\ub54c \\(L\\)\uc744 \uc218\uc5f4 \\(\\left\\{a_n \\right\\}\\)\uc758 \uadf9\ud55c<\/span>(limit) \ub610\ub294 \uadf9\ud55c\uac12<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uace0, \uc774\uac83\uc744 \uae30\ud638\ub85c
\n\\[\\lim_{n\\rightarrow \\infty} a_n = L\\]
\n\ub610\ub294
\n\\[a_n \\,\\, \\rightarrow \\,\\, L \\quad \\text{as} \\quad n\\,\\, \\rightarrow \\,\\, \\infty\\]
\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\n

\ubcf4\uae30 1.2.1.<\/span><\/p>\n

    \n
  1. \ub9cc\uc57d \\(a_n = \\displaystyle\\frac{1}{n}\\)\uc774\uba74 \\(\\left\\{a_n \\right\\}\\)\uc740 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc989 \\(\\displaystyle\\lim_{n\\rightarrow \\infty}\\frac{1}{n}=0\\)\uc774\ub2e4.<\/li>\n
  2. \ub9cc\uc57d \\(b_n = \\displaystyle\\frac{n+1}{n}\\)\uc774\uba74 \\(\\left\\{b_n \\right\\}\\)\uc740 \\(1\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc989 \\(\\displaystyle\\lim_{n\\rightarrow \\infty}\\frac{n+1}{n}=1\\)\uc774\ub2e4.<\/li>\n
  3. \ub9cc\uc57d \\(c_n = \\displaystyle\\frac{(-1)^n}{n}\\)\uc774\uba74 \\(\\left\\{c_n \\right\\}\\)\uc740 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc989 \\(\\displaystyle\\lim_{n\\rightarrow \\infty}\\frac{(-1)^n}{n}=0\\)\uc774\ub2e4.<\/li>\n
  4. \ub9cc\uc57d \\(s_n = 4\\)\uc774\uba74 \\(\\left\\{s_n \\right\\}\\)\uc740 \\(4\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc989 \\(\\displaystyle\\lim_{n\\rightarrow \\infty} 4=4\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

    <\/p>\n

    \ubc1c\uc0b0\ud558\ub294 \uc218\uc5f4<\/h2>\n

    \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc774 \uc5b4\ub5a0\ud55c \uac12\uc5d0\ub3c4 \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\uba74 \u201c\\(\\left\\{a_n\\right\\}\\)\uc774 \ubc1c\uc0b0<\/span>(diverge)\ud55c\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n

    \ub9cc\uc57d \\(n\\)\uc774 \ud55c\uc5c6\uc774 \ucee4\uc9c8 \ub54c \ud56d \\(a_n\\)\ub3c4 \ud55c\uc5c6\uc774 \ucee4\uc9c0\uba74, \u201c\\(\\left\\{a_n\\right\\}\\)\uc774 \uc591\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud55c\ub2e4\u201d\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\uac83\uc744 \uae30\ud638\ub85c
    \n\\[\\lim_{n\\rightarrow\\infty} a_n = \\infty\\]
    \n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \ub9cc\uc57d \\(n\\)\uc774 \ud55c\uc5c6\uc774 \ucee4\uc9c8 \ub54c \ud56d \\(-a_n\\)\ub3c4 \ud55c\uc5c6\uc774 \ucee4\uc9c0\uba74, \u201c\\(\\left\\{a_n\\right\\}\\)\uc774 \uc74c\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud55c\ub2e4\u201d\ub77c\uace0 \ub9d0\ud558\uace0, \uc774\uac83\uc744 \uae30\ud638\ub85c
    \n\\[\\lim_{n\\rightarrow\\infty} a_n = -\\infty\\]
    \n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

    \uc218\uc5f4 \\(\\left\\{a_n\\right\\}\\)\uc774 \uc218\ub834\ud558\uc9c0 \uc54a\uace0, \uc591\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud558\uc9c0 \uc54a\uace0, \uc74c\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud558\uc9c0\ub3c4 \uc54a\uc73c\uba74 \u201c\\(\\left\\{ a_n \\right\\}\\)\uc774 \uc9c4\ub3d9<\/span>(oscillate)\ud55c\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n

    \n

    \ubcf4\uae30 1.2.2.<\/span><\/p>\n

      \n
    1. \ub9cc\uc57d \\(a_n = n^2\\)\uc774\uba74 \\(\\left\\{a_n \\right\\}\\)\uc740 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud55c\ub2e4. \uc989 \\(\\displaystyle\\lim_{n\\rightarrow \\infty}n^2 = \\infty\\)\uc774\ub2e4.<\/li>\n
    2. \ub9cc\uc57d \\(b_n = -2^n\\)\uc774\uba74 \\(\\left\\{b_n \\right\\}\\)\uc740 \uc74c\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud55c\ub2e4. \uc989 \\(\\displaystyle\\lim_{n\\rightarrow \\infty}\\left(-2^n\\right)=-\\infty\\)\uc774\ub2e4.<\/li>\n
    3. \ub9cc\uc57d \\(c_n = (-1)^n\\)\uc774\uba74 \\(\\left\\{c_n \\right\\}\\)\uc740 \uc9c4\ub3d9\ud55c\ub2e4.<\/li>\n
    4. \ub9cc\uc57d \\(x_n = n+n(-1)^n\\)\uc774\uba74 \\(\\left\\{x_n \\right\\}\\)\uc740 \uc9c4\ub3d9\ud55c\ub2e4.<\/li>\n
    5. \ub9cc\uc57d \\(y_n = (-2)^n\\)\uc774\uba74 \\(\\left\\{y_n \\right\\}\\)\uc740 \uc9c4\ub3d9\ud55c\ub2e4.<\/li>\n
    6. \ub9cc\uc57d \\(z_n = (-n)^n\\)\uc774\uba74 \\(\\left\\{z_n \\right\\}\\)\uc740 \uc9c4\ub3d9\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

      <\/p>\n

      \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc5c4\ubc00\ud55c \uc815\uc758<\/h2>\n

      \uc774 \ucc45\uc5d0\uc11c \ub3c4\uc785\ud55c \uadf9\ud55c\uc758 \uac1c\ub150\uc740 \uc9c1\uad00\uc801\uc73c\ub85c \uc815\uc758\ub41c \uadf9\ud55c\uc774\ub2e4. \uc0ac\uc2e4 \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc5c4\ubc00\ud558\uac8c \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n

      \n

      \\(\\left\\{a_n\\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\uace0 \\(L\\)\uc774 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \uc591\uc218 \\(\\epsilon\\)\uc5d0 \ub300\ud558\uc5ec \uc591\uc758 \uc815\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc778 \ubaa8\ub4e0 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\left\\lvert a_n – L \\right\\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud558\uba74 \u201c\\(\\left\\{a_n\\right\\}\\)\uc774 \\(L\\)\uc5d0 \uc218\ub834<\/span>\ud55c\ub2e4\u201d\ub77c\uace0 \ub9d0\ud558\uace0, \\(L\\)\uc744 \\(\\left\\{ a_n \\right\\}\\)\uc758 \uadf9\ud55c<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.\n<\/p>\n<\/div>\n

      \uc704 \uc815\uc758\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uae30\ud638\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.
      \n\\[\\forall \\epsilon > 0 \\, \\exists N \\in \\mathbb{N} \\, \\forall n\\in\\mathbb{N} \\,:\\quad \\left[ n > N \\quad \\rightarrow \\quad \\left\\lvert a_n – L \\right\\rvert < \\epsilon \\right].\\]\n\uc5c4\ubc00\ud55c \uc815\uc758\ub97c \uc774\uc6a9\ud558\uc5ec \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uc99d\uba85\ud558\ub294 \ub2e4\uc74c \uc608\uc81c\ub97c \ubcf4\uc790.<\/p>\n

      \n

      \uc608\uc81c 1.2.3.<\/span>
      \n\\(b_n = \\displaystyle\\frac{n+1}{n}\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\displaystyle\\lim_{n\\rightarrow\\infty} b_n = 1\\)\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n

      \ud480\uc774.<\/span> \uc591\uc218 \\(\\epsilon\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc790\uc5f0\uc218\ub4e4\uc758 \ubaa8\uc784\uc740 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ubbc0\ub85c \\(N > 1\/\\epsilon\\)\uc778 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \\(n > N\\)\uc778 \ubaa8\ub4e0 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
      \n\\[\\left\\lvert b_n – 1 \\right\\rvert = \\left\\lvert \\frac{n+1}{n} – 1\\right\\rvert = \\frac{1}{n} < \\frac{1}{N} < \\epsilon.\\]\n\uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{b_n \\right\\}\\)\uc740 \\(1\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.\n<\/p>\n<\/div>\n

      \uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \uad00\ub828\ub41c \ubaa8\ub4e0 \uc131\uc9c8\uc740 \\(\\epsilon-N\\) \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc99d\uba85\ud560 \uc218 \uc788\ub2e4. \uc774 \ucc45\uc5d0\uc11c\ub294 \\(\\epsilon-N\\) \uc815\uc758\ub97c \uc0ac\uc6a9\ud55c \uc99d\uba85\uc744 \ub354 \uc0c1\uc138\ud558\uac8c \ub2e4\ub8e8\uc9c0\ub294 \uc54a\uaca0\ub2e4.<\/p>\n


      \n<\/p>\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