{"id":6679,"date":"2021-07-20T23:50:47","date_gmt":"2021-07-20T14:50:47","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6679"},"modified":"2021-08-30T12:46:42","modified_gmt":"2021-08-30T03:46:42","slug":"limit-of-a-vector-sequence","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/limit-of-a-vector-sequence\/","title":{"rendered":"\ubca1\ud130\uc218\uc5f4\uc758 \uadf9\ud55c"},"content":{"rendered":"
\n

\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 1\uc7a5 7\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.  (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n

\\(\\mathbb{R}^d\\)\uac00 \uc720\ud074\ub9ac\ub4dc \uacf5\uac04\uc774\uace0 \\(d\\)\uac00 \\(2\\) \uc774\uc0c1\uc778 \uc815\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n

\\(\\mathbf{v} = \\left(v_1 ,\\, v_2 ,\\, \\cdots ,\\, v_d \\right) \\in\\mathbb{R}^d \\)\uc77c \ub54c \\(\\mathbf{v}\\)\uc758 \ub178\ub984<\/span>(norm)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
\n\\[\\left\\lvert \\mathbf{v} \\right\\rvert = \\sqrt{v_1 ^2 + v_2 ^2 + \\cdots + v_d ^2} .\\]
\n\\(\\mathbf{u}\\)\uc640 \\(\\mathbf{v}\\)\uac00 \\(\\mathbb{R}^d\\)\uc758 \uc810\uc77c \ub54c,
\n\\[\\left\\lvert \\mathbf{u} – \\mathbf{v} \\right\\rvert\\]
\n\ub97c \\(\\mathbf{u}\\)\uc640 \\(\\mathbf{v}\\) \uc0ac\uc774\uc758 \uac70\ub9ac<\/span>(distance)\ub77c\uace0 \uc815\uc758\ud55c\ub2e4.<\/p>\n

\uc720\ud074\ub9ac\ub4dc \uacf5\uac04\uc5d0\uc11c \uc810 \uc0ac\uc774\uc758 \uac70\ub9ac\ub97c \uc815\uc758\ud558\uc600\uc73c\ubbc0\ub85c, \uc774\uc81c \uc720\ud074\ub9ac\ub4dc \uacf5\uac04\uc5d0\uc11c\uc758 \uadf9\ud55c\uc744 \ub17c\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\\(\\left\\{ \\mathbf{a}_n\\right\\}\\)\uc774 \ubca1\ud130\uc218\uc5f4<\/span>(vector sequence)\uc774\ub77c\uace0 \ud558\uc790. \uc989 \uac01 \ud56d \\(\\mathbf{a}_n\\)\uc774 \\(\\mathbb{R}^d\\)\uc5d0 \uc18d\ud55c\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \ubca1\ud130 \\(\\mathbf{L} \\in \\mathbb{R}^d\\)\uc774 \uc874\uc7ac\ud558\uc5ec
\n\\[\\lim_{n\\rightarrow\\infty} \\left\\lvert \\mathbf{a}_n – \\mathbf{L} \\right\\rvert = 0\\]
\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \u201c\\(\\left\\{ \\mathbf{a}_n\\right\\}\\)\uc774 \\(\\mathbf{L}\\)\uc5d0 \uc218\ub834<\/span>\ud55c\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc774\ub54c \\(\\mathbf{L}\\)\uc744 \\(\\left\\{ \\mathbf{a}_n\\right\\}\\)\uc758 \uadf9\ud55c<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\uac83\uc744 \uae30\ud638\ub85c
\n\\[\\lim_{n\\rightarrow\\infty}\\mathbf{a}_n = \\mathbf{L}\\]
\n\ub610\ub294
\n\\[\\mathbf{a}_n \\,\\rightarrow\\,\\mathbf{L} \\quad\\text{as}\\quad n\\,\\rightarrow\\,\\infty\\]
\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\n

\ubcf4\uae30 1.7.1.<\/span><\/p>\n

    \n
  1. \\(\\mathbf{a}_n = \\left( \\frac{1}{n} ,\\, \\frac{n}{n+1}\\right)\\)\uc774\uace0 \\(\\mathbf{L} = (0,\\,1)\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(n\\rightarrow\\infty\\)\uc77c \ub54c
    \n\\[\\left\\lvert \\mathbf{a}_n – \\mathbf{L} \\right\\rvert
    \n= \\sqrt{\\frac{1}{n^2} + \\frac{1}{(n+1)^2}}
    \n\\le \\sqrt{ \\frac{1}{n^2} + \\frac{1}{n^2} }
    \n= \\frac{\\sqrt{2}}{n}
    \n\\,\\rightarrow\\,0\\]
    \n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{ \\mathbf{a}_n \\right\\}\\)\uc740 \\((0,\\,1)\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/li>\n
  2. \\(\\mathbf{b}_n = \\left(\\frac{1}{n} ,\\, 2n \\right)\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\mathbf{L} = \\left( L_1 ,\\, L_2 \\right)\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc9c4 \ubca1\ud130\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(n\\rightarrow\\infty\\)\uc77c \ub54c
    \n\\[\\left\\lvert \\mathbf{b}_n – \\mathbf{L} \\right\\rvert
    \n= \\sqrt{ \\left( \\frac{1}{n} – L_1 \\right)^2 + \\left( 2n-L_2 \\right)^2 }
    \n\\,\\rightarrow\\, \\infty\\]
    \n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{ \\mathbf{b}_n \\right\\}\\)\uc740 \uc5b4\ub5a0\ud55c \ubca1\ud130 \\(\\mathbf{L}\\)\uc5d0\ub3c4 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

    \ub2e4\uc74c \uc815\ub9ac\ub294 \ubca1\ud130\uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uacc4\uc0b0\ud560 \ub54c \uc720\uc6a9\ud558\uac8c \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 1.7.1.<\/span><\/p>\n

    \\(\\left\\{ x_n \\right\\},\\) \\(\\left\\{ y_n \\right\\},\\) \\(\\left\\{ z_n \\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\uace0, \\(\\left\\{\\mathbf{a}_n\\right\\}\\)\uc774
    \n\\[\\mathbf{a}_n = \\left( x_n ,\\, y_n ,\\, z_n \\right)\\]
    \n\uc73c\ub85c \uc815\uc758\ub41c \ubca1\ud130\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\left\\{\\mathbf{a}_n\\right\\}\\)\uc774 \\((x,\\,y,\\,z)\\)\uc5d0 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\left\\{ x_n \\right\\},\\) \\(\\left\\{ y_n \\right\\},\\) \\(\\left\\{ z_n \\right\\}\\)\uc774 \uac01\uac01 \\(x,\\) \\(y,\\) \\(z\\)\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n

    \n

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

    \ubd80\ub4f1\uc2dd
    \n\\[ \\begin{gather}
    \n\\left\\lvert x_n – x \\right\\rvert \\le \\left\\lvert \\mathbf{a}_n – \\mathbf{L} \\right\\rvert , \\\\[5pt]
    \n\\left\\lvert y_n – y \\right\\rvert \\le \\left\\lvert \\mathbf{a}_n – \\mathbf{L} \\right\\rvert , \\\\[5pt]
    \n\\left\\lvert z_n – z \\right\\rvert \\le \\left\\lvert \\mathbf{a}_n – \\mathbf{L} \\right\\rvert
    \n\\end{gather} \\]
    \n\uacfc
    \n\\[ \\left\\lvert \\mathbf{a}_n – \\mathbf{L} \\right\\rvert \\le \\left\\lvert x_n – x \\right\\rvert + \\left\\lvert y_n – y \\right\\rvert + \\left\\lvert z_n – z \\right\\rvert \\]
    \n\ub97c \uc774\uc6a9\ud558\uba74 \uc99d\uba85\ub41c\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \n

    \ubcf4\uae30 1.7.2.<\/span><\/p>\n

      \n
    1. \\(\\mathbf{a}_n = \\left( \\frac{1}{n} ,\\, \\frac{n}{n+1}\\right)\\)\uc774\uba74, \\(n\\rightarrow\\infty\\)\uc77c \ub54c
      \n\\[\\frac{1}{n} \\,\\rightarrow\\,0 \\quad\\text{and}\\quad \\frac{n}{n+1} \\,\\rightarrow\\,1\\]
      \n\uc774\ubbc0\ub85c
      \n\\(\\left\\{ \\mathbf{a}_n \\right\\}\\)\uc740 \\((0,\\,1)\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/li>\n
    2. \\(\\mathbf{b}_n = \\left(\\frac{1}{n} ,\\, 2n \\right)\\)\uc774\uba74, \\(n\\rightarrow\\infty\\)\uc77c \ub54c
      \n\\[2n \\,\\rightarrow\\,\\infty\\]
      \n\uc774\ubbc0\ub85c \\(\\left\\{\\mathbf{b}_n\\right\\}\\)\uc740 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

      \uc784\uc758\uc758 \ubcf5\uc18c\uc218 \\(z\\)\ub294 \uc2e4\uc218 \\(a,\\) \\(b\\)\ub97c \uc0ac\uc6a9\ud558\uc5ec \\(z = a+b\\boldsymbol{i}\\)\uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc73c\uba70, \ud558\ub098\uc758 \ubcf5\uc18c\uc218 \\(z\\)\uc5d0 \uc774\uc640 \uac19\uc740 \uaf34\ub85c \ub300\uc751\ub418\ub294 \uc2e4\uc218 \\(a,\\) \\(b\\)\ub294 \uac01\uac01 \uc720\uc77c\ud558\ub2e4. \ub530\ub77c\uc11c \ubcf5\uc18c\uc218 \\(z = a+b\\boldsymbol{i}\\)\ub97c \uc21c\uc11c\uc30d \\((a,\\,b)\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\uba74 \\(\\mathbb{C}\\)\ub294 2\ucc28\uc6d0 \uc720\ud074\ub9ac\ub4dc \uacf5\uac04\uacfc \uac19\ub2e4. \uc774\uc640 \uac19\uc740 \uad00\uc810\uc5d0\uc11c \uac01 \ubcf5\uc18c\uc218\ub294 2\ucc28\uc6d0 \ubca1\ud130\uc774\uba70, \ubcf5\uc18c\uc218\uc5f4\uc758 \uadf9\ud55c\uc740 \ubca1\ud130\uc218\uc5f4\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc815\uc758\ub41c\ub2e4.<\/p>\n

      \\(\\left\\{z_n\\right\\}\\)\uc774 \ubcf5\uc18c\uc218\uc5f4\uc774\uace0 \\(z_n = a_n + b_n \\boldsymbol{i}\\)\uc774\uba70 \\(\\left\\{a_n \\right\\}\\)\uacfc \\(\\left\\{b_n\\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\left\\{ z_n \\right\\}\\)\uc774 \\(z = a+b\\boldsymbol{i}\\)\uc5d0 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\left\\{ a_n \\right\\}\\)\uacfc \\(\\left\\{ b_n \\right\\}\\)\uc774 \uac01\uac01 \\(a\\)\uc640 \\(b\\)\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4. \uc5ec\uae30\uc11c \\(z\\)\ub97c \uc218\uc5f4 \\(\\left\\{z_n\\right\\}\\)\uc758 \uadf9\ud55c<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uba70, \uc774\uac83\uc744 \uae30\ud638\ub85c
      \n\\[\\lim_{n\\rightarrow\\infty} z_n = z\\]
      \n\ub610\ub294
      \n\\[z_n \\,\\rightarrow\\,z \\quad\\text{as}\\quad n\\,\\rightarrow\\,\\infty\\]
      \n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

      \n

      \ubcf4\uae30 1.7.3.<\/span>
      \n\\[z_n = \\frac{4n}{2n+1} + \\frac{3n+1}{n-1}\\boldsymbol{i}\\]
      \n\uc774\uba74
      \n\\[\\lim_{n\\rightarrow\\infty} z_n = 2 + 3 \\boldsymbol{i}\\]
      \n\uc774\ub2e4.\n<\/p>\n<\/div>\n

      \uc720\ud074\ub9ac\ub4dc \uacf5\uac04\uc5d0\uc11c \uc815\uc758\ub41c \ubca1\ud130\uc218\uc5f4\uc758 \uc5ec\ub7ec \uac00\uc9c0 \uc131\uc9c8\uc740 \uc2e4\uc218\uc5f4\uc758 \uc131\uc9c8\ub85c\ubd80\ud130 \uc720\ub3c4\ub41c\ub2e4. \uadf8\ub7ec\ub098 \uc774 \ucc45\uc5d0\uc11c\ub294 \ubca1\ud130\uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \uad00\ub828\ub41c \uc774\ub860\uc744 \ub354 \uae4a\uc774 \ub2e4\ub8e8\uc9c0\ub294 \uc54a\uaca0\ub2e4.<\/p>\n


      \n<\/p>\n

      <\/p>\n

      <\/p>\n

      \n