{"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":"2026-09-27T17:18:05","modified_gmt":"2026-09-27T08:18:05","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 \\(d\\)\ucc28\uc6d0 \uc720\ud074\ub9ac\ub4dc \uacf5\uac04\uc774\uace0 \\(d\\)\uac00 \uc591\uc758 \uc815\uc218\ub77c\uace0 \ud558\uc790. 0\uc7a5 8\uc808\uc5d0\uc11c \ubca1\ud130
\n\\[
\n\\mathbf{v}
\n=
\n(v_1,\\,v_2,\\,\\ldots,\\,v_d)
\n\\in\\mathbb{R}^d
\n\\]
\n\uc758 \ub178\ub984\uc744
\n\\[
\n\\lVert\\mathbf{v}\\rVert
\n=
\n\\sqrt{v_1^2+v_2^2+\\cdots+v_d^2}
\n\\]
\n\ub85c \uc815\uc758\ud558\uc600\ub2e4.<\/p>\n

\\(\\mathbf{u},\\mathbf{v}\\in\\mathbb{R}^d\\)\uc77c \ub54c
\n\\[
\n\\lVert\\mathbf{u}-\\mathbf{v}\\rVert
\n\\]
\n\ub97c \\(\\mathbf{u}\\)\uc640 \\(\\mathbf{v}\\) \uc0ac\uc774\uc758 \uac70\ub9ac<\/span>(distance)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\uc81c \uc774 \uac70\ub9ac\ub97c \uc774\uc6a9\ud558\uc5ec \ubca1\ud130\uc218\uc5f4\uc758 \uadf9\ud55c\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n

<\/p>\n

\ubca1\ud130\uc218\uc5f4\uc758 \uadf9\ud55c<\/h2>\n

\\(\\left\\{\\mathbf{a}_n\\right\\}\\)\uc774 \ubca1\ud130\uc218\uc5f4<\/span>(vector sequence), \uc989 \uac01 \ud56d \\(\\mathbf{a}_n\\)\uc774 \\(\\mathbb{R}^d\\)\uc5d0 \uc18d\ud558\ub294 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \ubca1\ud130 \\(\\mathbf{L}\\in\\mathbb{R}^d\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[
\n\\lim_{n\\rightarrow\\infty}
\n\\lVert\\mathbf{a}_n-\\mathbf{L}\\rVert
\n=
\n0
\n\\]
\n\uc774\uba74 \\(\\left\\{\\mathbf{a}_n\\right\\}\\)\uc774 \\(\\mathbf{L}\\)\uc5d0 \uc218\ub834<\/span>\ud55c\ub2e4\uace0 \ub9d0\ud55c\ub2e4. \uc774\ub54c \\(\\mathbf{L}\\)\uc744 \\(\\left\\{\\mathbf{a}_n\\right\\}\\)\uc758 \uadf9\ud55c<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uace0
\n\\[
\n\\lim_{n\\rightarrow\\infty}\\mathbf{a}_n
\n=
\n\\mathbf{L}
\n\\]
\n\ub610\ub294
\n\\[
\n\\mathbf{a}_n\\rightarrow\\mathbf{L}
\n\\quad\\text{as}\\quad
\nn\\rightarrow\\infty
\n\\]
\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\uc2e4\uc218\uc5f4\uc758 \uadf9\ud55c\uc5d0 \ub300\ud55c \\(\\epsilon\\)-\\(N\\) \uc815\uc758\ub97c \uc774\uc6a9\ud558\uba74 \uc704 \uc815\uc758\ub294 \ub2e4\uc74c \uc870\uac74\uacfc \ub3d9\uce58\uc774\ub2e4. \uc784\uc758\uc758 \uc591\uc218 \\(\\epsilon\\)\uc5d0 \ub300\ud558\uc5ec \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \ucca8\uc790 \\(n\\)\uc5d0\uc11c
\n\\[
\n\\lVert\\mathbf{a}_n-\\mathbf{L}\\rVert < \\epsilon\n\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

\\(d=1\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(\\mathbb{R}^1\\)\uc744 \\(\\mathbb{R}\\)\uacfc \ub3d9\uc77c\uc2dc\ud560 \uc218 \uc788\uace0
\n\\[
\n\\lVert a_n-L\\rVert=|a_n-L|
\n\\]
\n\uc774\ubbc0\ub85c \uc704 \uc815\uc758\ub294 \uc2e4\uc218\uc5f4\uc758 \uc218\ub834 \uc815\uc758\uc640 \uc77c\uce58\ud55c\ub2e4.<\/p>\n

\n

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

    \n
  1. \\[
    \n\\mathbf{a}_n
    \n=
    \n\\left(
    \n\\frac{1}{n},\\,
    \n\\frac{n}{n+1}
    \n\\right)
    \n\\]
    \n\uc774\uace0
    \n\\[
    \n\\mathbf{L}=(0,\\,1)
    \n\\]
    \n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
    \n\\[
    \n\\begin{aligned}
    \n\\lVert\\mathbf{a}_n-\\mathbf{L}\\rVert
    \n&=
    \n\\sqrt{
    \n\\frac{1}{n^2}
    \n+
    \n\\left(
    \n\\frac{n}{n+1}-1
    \n\\right)^2
    \n}\\\\
    \n&=
    \n\\sqrt{
    \n\\frac{1}{n^2}
    \n+
    \n\\frac{1}{(n+1)^2}
    \n}\\\\
    \n&\\le
    \n\\frac{\\sqrt{2}}{n}.
    \n\\end{aligned}
    \n\\]
    \n\uadf8\ub7f0\ub370
    \n\\[
    \n\\frac{\\sqrt{2}}{n}\\rightarrow0
    \n\\]
    \n\uc774\ubbc0\ub85c \uc870\uc784 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec
    \n\\[
    \n\\lVert\\mathbf{a}_n-\\mathbf{L}\\rVert\\rightarrow0.
    \n\\]
    \n\ub530\ub77c\uc11c
    \n\\[
    \n\\mathbf{a}_n\\rightarrow(0,\\,1).
    \n\\]\n<\/li>\n
  2. \\[
    \n\\mathbf{b}_n
    \n=
    \n\\left(
    \n\\frac{1}{n},\\,2n
    \n\\right)
    \n\\]
    \n\uc774\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \ubca1\ud130
    \n\\[
    \n\\mathbf{L}=(L_1,\\,L_2)
    \n\\]
    \n\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n\\begin{aligned}
    \n\\lVert\\mathbf{b}_n-\\mathbf{L}\\rVert
    \n&=
    \n\\sqrt{
    \n\\left(
    \n\\frac1n-L_1
    \n\\right)^2
    \n+
    \n(2n-L_2)^2
    \n}\\\\
    \n&\\ge
    \n|2n-L_2|.
    \n\\end{aligned}
    \n\\]
    \n\ub610\ud55c
    \n\\[
    \n|2n-L_2|
    \n\\ge
    \n2n-|L_2|
    \n\\rightarrow\\infty.
    \n\\]
    \n\ub530\ub77c\uc11c
    \n\\[
    \n\\lVert\\mathbf{b}_n-\\mathbf{L}\\rVert\\rightarrow\\infty.
    \n\\]
    \n\uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{\\mathbf{b}_n\\right\\}\\)\uc740 \uc5b4\ub5a0\ud55c \ubca1\ud130\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 \uac01 \uc131\ubd84\uc758 \uadf9\ud55c\uc73c\ub85c \uacc4\uc0b0\ud560 \uc218 \uc788\uc74c\uc744 \ubcf4\uc5ec \uc900\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 1.7.1. (\uc131\ubd84\ubcc4 \uc218\ub834)<\/span><\/p>\n

    \n\\[
    \n\\mathbf{a}_n
    \n=
    \n\\left(
    \na_n^{(1)},\\,
    \na_n^{(2)},\\,
    \n\\ldots,\\,
    \na_n^{(d)}
    \n\\right)
    \n\\]
    \n\uc774\uace0
    \n\\[
    \n\\mathbf{L}
    \n=
    \n(L_1,\\,L_2,\\,\\ldots,\\,L_d)
    \n\\]
    \n\ub77c\uace0 \ud558\uc790. \uc774\ub54c
    \n\\[
    \n\\mathbf{a}_n\\rightarrow\\mathbf{L}
    \n\\]
    \n\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ubaa8\ub4e0 \\(j=1,2,\\ldots,d\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \na_n^{(j)}\\rightarrow L_j
    \n\\]
    \n\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

    \n

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

    \uba3c\uc800
    \n\\[
    \n\\mathbf{a}_n\\rightarrow\\mathbf{L}
    \n\\]
    \n\uc774\ub77c\uace0 \ud558\uc790. \uac01 \\(j=1,\\ldots,d\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n\\left|
    \na_n^{(j)}-L_j
    \n\\right|
    \n\\le
    \n\\lVert\\mathbf{a}_n-\\mathbf{L}\\rVert.
    \n\\]
    \n\uc624\ub978\ucabd\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \uc870\uc784 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec
    \n\\[
    \na_n^{(j)}\\rightarrow L_j.
    \n\\]\n<\/p>\n

    \uac70\uafb8\ub85c \ubaa8\ub4e0 \\(j=1,\\ldots,d\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \na_n^{(j)}\\rightarrow L_j
    \n\\]
    \n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
    \n\\[
    \n\\left(
    \na_n^{(j)}-L_j
    \n\\right)^2
    \n\\rightarrow0
    \n\\]
    \n\uc774\ubbc0\ub85c \uc218\uc5f4\uc758 \uadf9\ud55c\uc5d0 \uad00\ud55c \uc0ac\uce59\uc5f0\uc0b0 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec
    \n\\[
    \n\\sum_{j=1}^{d}
    \n\\left(
    \na_n^{(j)}-L_j
    \n\\right)^2
    \n\\rightarrow0.
    \n\\]
    \n\ub530\ub77c\uc11c
    \n\\[
    \n\\begin{aligned}
    \n\\lVert\\mathbf{a}_n-\\mathbf{L}\\rVert
    \n&=
    \n\\sqrt{
    \n\\sum_{j=1}^{d}
    \n\\left(
    \na_n^{(j)}-L_j
    \n\\right)^2
    \n}\\\\
    \n&\\rightarrow0.
    \n\\end{aligned}
    \n\\]
    \n\uadf8\ub7ec\ubbc0\ub85c
    \n\\[
    \n\\mathbf{a}_n\\rightarrow\\mathbf{L}.
    \n\\]
    \n<\/span><\/p>\n<\/div>\n

    \n

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

      \n
    1. \\[
      \n\\mathbf{a}_n
      \n=
      \n\\left(
      \n\\frac{1}{n},\\,
      \n\\frac{n}{n+1}
      \n\\right)
      \n\\]
      \n\uc774\uba74
      \n\\[
      \n\\frac1n\\rightarrow0,
      \n\\qquad
      \n\\frac{n}{n+1}\\rightarrow1.
      \n\\]
      \n\ub530\ub77c\uc11c \uc815\ub9ac 1.7.1\uc5d0 \uc758\ud558\uc5ec
      \n\\[
      \n\\mathbf{a}_n\\rightarrow(0,\\,1).
      \n\\]\n<\/li>\n
    2. \\[
      \n\\mathbf{b}_n
      \n=
      \n\\left(
      \n\\frac1n,\\,2n
      \n\\right)
      \n\\]
      \n\uc774\uba74 \ub450 \ubc88\uc9f8 \uc131\ubd84 \\(2n\\)\uc774 \uc2e4\uc218\ub85c \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \ub530\ub77c\uc11c \uc815\ub9ac 1.7.1\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{\\mathbf{b}_n\\right\\}\\)\uc740 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

      <\/p>\n

      \ubcf5\uc18c\uc218\uc5f4\uc758 \uadf9\ud55c<\/h2>\n

      \uc784\uc758\uc758 \ubcf5\uc18c\uc218 \\(z\\)\ub294 \uc720\uc77c\ud55c \uc2e4\uc218 \\(a,b\\)\ub97c \uc0ac\uc6a9\ud558\uc5ec
      \n\\[
      \nz=a+b\\boldsymbol{i}
      \n\\]
      \n\uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \ub300\uc751
      \n\\[
      \na+b\\boldsymbol{i}
      \n\\longleftrightarrow
      \n(a,\\,b)
      \n\\]
      \n\ub97c \uc774\uc6a9\ud558\uc5ec \\(\\mathbb{C}\\)\ub97c \\(\\mathbb{R}^2\\)\uc640 \uc790\uc5f0\uc2a4\ub7fd\uac8c \ub3d9\uc77c\uc2dc\ud560 \uc218 \uc788\ub2e4. \uc815\ud655\ud788 \ub9d0\ud558\uba74 \ubcf5\uc18c\uc218\uc758 \ub367\uc148\uacfc \uc2e4\uc218 \uc2a4\uce7c\ub77c\ubc30\ub97c \uace0\ub824\ud560 \ub54c \\(\\mathbb{C}\\)\ub294 \\(\\mathbb{R}^2\\)\uc640 \uac19\uc740 \uad6c\uc870\uc758 \uc2e4\uc218 \ubca1\ud130\uacf5\uac04\uc774\ub2e4.<\/p>\n

      \ubcf5\uc18c\uc218
      \n\\[
      \nz=a+b\\boldsymbol{i}
      \n\\]
      \n\uc758 \uc808\ub313\uac12<\/span>(absolute value) \ub610\ub294 \ubaa8\ub4c8\ub7ec\uc2a4<\/span>(modulus)\ub97c
      \n\\[
      \n|z|
      \n=
      \n\\sqrt{a^2+b^2}
      \n\\]
      \n\ub85c \uc815\uc758\ud55c\ub2e4. \uc774 \uac12\uc740 \\(z\\)\uc5d0 \ub300\uc751\ud558\ub294 \ubca1\ud130 \\((a,b)\\)\uc758 \uc720\ud074\ub9ac\ub4dc \ub178\ub984\uacfc \uac19\ub2e4. \ub530\ub77c\uc11c \ub450 \ubcf5\uc18c\uc218 \\(z,w\\) \uc0ac\uc774\uc758 \uac70\ub9ac\ub294
      \n\\[
      \n|z-w|
      \n\\]
      \n\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n

      \\(\\left\\{z_n\\right\\}\\)\uc774 \ubcf5\uc18c\uc218\uc5f4\uc774\uace0 \\(z\\in\\mathbb{C}\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d
      \n\\[
      \n|z_n-z|\\rightarrow0
      \n\\]
      \n\uc774\uba74 \\(\\left\\{z_n\\right\\}\\)\uc774 \\(z\\)\uc5d0 \uc218\ub834<\/span>\ud55c\ub2e4\uace0 \ub9d0\ud558\uace0
      \n\\[
      \n\\lim_{n\\rightarrow\\infty}z_n=z
      \n\\]
      \n\ub610\ub294
      \n\\[
      \nz_n\\rightarrow z
      \n\\quad\\text{as}\\quad
      \nn\\rightarrow\\infty
      \n\\]
      \n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

      \uc774\uc81c
      \n\\[
      \nz_n=a_n+b_n\\boldsymbol{i},
      \n\\qquad
      \nz=a+b\\boldsymbol{i}
      \n\\]
      \n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
      \n\\[
      \n|z_n-z|
      \n=
      \n\\sqrt{
      \n(a_n-a)^2+(b_n-b)^2
      \n}.
      \n\\]
      \n\ub530\ub77c\uc11c \uc815\ub9ac 1.7.1\uc5d0\uc11c \\(d=2\\)\uc778 \uacbd\uc6b0\ub97c \uc801\uc6a9\ud558\uba74
      \n\\[
      \nz_n\\rightarrow z
      \n\\]
      \n\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740
      \n\\[
      \na_n\\rightarrow a
      \n\\qquad\\text{and}\\qquad
      \nb_n\\rightarrow b
      \n\\]
      \n\uc778 \uac83\uc774\ub2e4.<\/p>\n

      \n

      \ubcf4\uae30 1.7.3.<\/span>
      \n\\(n\\ge2\\)\uc77c \ub54c
      \n\\[
      \nz_n
      \n=
      \n\\frac{4n}{2n+1}
      \n+
      \n\\frac{3n+1}{n-1}\\boldsymbol{i}
      \n\\]
      \n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
      \n\\[
      \n\\frac{4n}{2n+1}\\rightarrow2
      \n\\]
      \n\uc774\uace0
      \n\\[
      \n\\frac{3n+1}{n-1}\\rightarrow3
      \n\\]
      \n\uc774\ubbc0\ub85c
      \n\\[
      \n\\lim_{n\\rightarrow\\infty}z_n
      \n=
      \n2+3\\boldsymbol{i}.
      \n\\]\n<\/p>\n<\/div>\n

      \uc774\ucc98\ub7fc \uc720\ud074\ub9ac\ub4dc \uacf5\uac04\uc758 \ubca1\ud130\uc218\uc5f4\uacfc \ubcf5\uc18c\uc218\uc5f4\uc758 \uc218\ub834\uc740 \uac01 \uc131\ubd84\uc744 \uc774\ub8e8\ub294 \uc2e4\uc218\uc5f4\uc758 \uc218\ub834\uc73c\ub85c \ud658\uc6d0\ud560 \uc218 \uc788\ub2e4. \uc774\ud6c4 \ubca1\ud130\uac12 \ud568\uc218\ub098 \uc5ec\ub7ec \ubcc0\uc218\ub97c \uac00\uc9c4 \ud568\uc218\uc758 \uadf9\ud55c\uc5d0\uc11c\ub3c4 \uac19\uc740 \uad00\uc810\uc774 \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n


      \n<\/p>\n

      <\/p>\n

      <\/p>\n

      \n