{"id":9493,"date":"2025-10-20T18:59:04","date_gmt":"2025-10-20T09:59:04","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9493"},"modified":"2025-10-21T16:09:05","modified_gmt":"2025-10-21T07:09:05","slug":"ch08-real-analytic-functions","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-analysis\/ch08-real-analytic-functions\/","title":{"rendered":"\uc2e4\ud574\uc11d\uc801 \ud568\uc218"},"content":{"rendered":"
<\/p>\n

<\/p>\n

\uc774 \uc7a5\uc5d0\uc11c\ub294 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf4\uace0, \ud568\uc218\ub97c \ud14c\uc77c\ub7ec \uae09\uc218\ub85c \ud45c\ud604\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

\ud568\uc218\uc5f4\uc758 \uade0\ub4f1\uc218\ub834<\/h3>\n

\uac01 \ud56d\uc774 \ud568\uc218\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc218\uc5f4\uc744 \ud568\uc218\uc5f4<\/span>(sequence of functions)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\{f_n\\}\\)\uc774 \ud568\uc218\uc5f4\uc774\uace0 \\(f\\)\uac00 \ud568\uc218\uc774\uba70 \uc784\uc758\uc758 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\lim_{n\\rightarrow\\infty} f_n (x) = f(x)\\]
\n\uc77c \ub54c, “\\(I\\)\uc5d0\uc11c \\(\\left\\{ f_n \\right\\}\\)\uc774 \\(f\\)\uc5d0 \uc218\ub834\ud55c\ub2e4”\ub77c\uace0 \ub9d0\ud558\uace0, \\(f\\)\ub97c \\(\\left\\{ f_n \\right\\}\\)\uc758 \uc810\ubcc4\uadf9\ud55c\ud568\uc218<\/span>(pointwise limit function) \ub610\ub294 \uac04\ub2e8\ud788 \uadf9\ud55c\ud568\uc218<\/span>(limit function)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\uac83\uc744 \\(f_n \\rightarrow f\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\n

\ubb38\uc81c 8.1.<\/span>
\n\\(f_n (x)=x^n\\)\uc77c \ub54c \\([0,\\,1]\\)\uc5d0\uc11c \ud568\uc218\uc5f4 \\(\\left\\{ f_n \\right\\}\\)\uc758 \uadf9\ud55c\ud568\uc218\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

\n

\ubb38\uc81c 8.2.<\/span>
\n\uc218\uc5f4 \\(\\left\\{ x_n \\right\\}\\)\uc774 \ubaa8\ub4e0 \uc720\ub9ac\uc218\ub97c \ud55c \ubc88\uc529 \ucde8\ud558\ub294 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \ud568\uc218 \\(f_n\\)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.
\n\\[f_n (x) = \\begin{cases}1&\\,\\,\\text{if }\\,x\\in\\left\\{x_1,\\,x_2,\\,\\cdots,\\,x_n\\right\\},\\\\[6pt] 0&\\,\\,\\text{if }\\,x\\notin\\left\\{x_1,\\,x_2,\\,\\cdots,\\,x_n\\right\\}.\\end{cases}\\]
\n\uc774\ub54c \ud568\uc218\uc5f4 \\(\\left\\{ f_n \\right\\}\\)\uc758 \uadf9\ud55c\ud568\uc218\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

\ud568\uc218\uc5f4\uc758 \uc218\ub834\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc57d\uc810\uc774 \uc788\ub2e4. \uc989 \ud568\uc218\uc5f4 \\(\\left\\{ f_n \\right\\}\\)\uc774 \\(f\\)\uc5d0 \uc810\ubcc4\uc218\ub834\ud560 \ub54c \ub2e4\uc74c\uacfc \uac19\uc740 \uacbd\uc6b0\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n

    \n
  • \ubaa8\ub4e0 \\(f_n\\)\uc774 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uc9c0\ub9cc \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \uc544\ub2d0 \uc218 \uc788\ub2e4.<\/li>\n
  • \ubaa8\ub4e0 \\(f_n\\)\uc774 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0\ub9cc \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \ubd88\uac00\ub2a5\ud560 \uc218 \uc788\ub2e4.<\/li>\n
  • \ubaa8\ub4e0 \\(f_n\\)\uc774 \ubbf8\ubd84 \uac00\ub2a5\ud558\uc9c0\ub9cc \\(f\\)\ub294 \ubbf8\ubd84 \ubd88\uac00\ub2a5\ud560 \uc218 \uc788\ub2e4.<\/li>\n<\/ul>\n

    \uc774\ub7ec\ud55c \ub2e8\uc810\uc744 \ubcf4\uc644\ud558\uae30 \uc704\ud574 \uade0\ub4f1\uc218\ub834 \uac1c\ub150\uc774 \ud544\uc694\ud558\ub2e4. \ub9cc\uc57d
    \n\\[\\lVert f \\rVert_\\infty = \\sup \\left\\{ \\lvert f(x) \\rvert \\mid x\\in I \\right\\}\\]
    \n\ub77c\uace0 \uc815\uc758\ub41c \ub178\ub984 \\(\\lVert \\cdot \\rVert_\\infty\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[\\lim_{n\\rightarrow\\infty} \\left\\lVert f_n – f \\right\\rVert_\\infty = 0\\]
    \n\uc774 \uc131\ub9bd\ud558\uba74, “\\(I\\)\uc5d0\uc11c \\(\\left\\{ f_n \\right\\}\\)\uc774 \\(f\\)\uc5d0 \uade0\ub4f1\uc218\ub834<\/span>\ud55c\ub2e4(converges uniformly)”\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc774\uac83\uc744 \\(f_n \\rightrightarrows f\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

    \uc815\uc758\uc5d0 \uc758\ud558\uc5ec, \\(I\\)\uc5d0\uc11c \\(\\left\\{ f_n \\right\\}\\)\uc774 \\(f\\)\uc5d0 \uade0\ub4f1\uc218\ub834\ud558\uba74 \\(\\left\\{ f_n \\right\\}\\)\uc740 \\(f\\)\uc5d0 \uc810\ubcc4\uc218\ub834\ud55c\ub2e4.<\/p>\n

    \n

    \ubb38\uc81c 8.3.<\/span>
    \n\uad6c\uac04 \\((0,\\,1]\\)\uc5d0\uc11c \uc77c\ubc18\ud56d\uc774 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\ub294 \ud568\uc218\uc5f4 \\(\\left\\{ f_n \\right\\}\\)\uc774 \uade0\ub4f1\uc218\ub834\ud558\ub294\uc9c0 \ud310\uc815\ud558\uc2dc\uc624.<\/p>\n

      \n
    1. \\(f_n(x) = \\frac{1}{n+x}\\)<\/li>\n
    2. \\(f_n(x) = \\frac{\\sin nx}{n}\\)<\/li>\n
    3. \\(f_n(x) = nx(1-x)^n\\)<\/li>\n
    4. \\(f_n(x) = \\frac{1}{nx+1}\\)<\/li>\n
    5. \\(f_n(x) = \\frac{1}{x} + \\frac{nx}{e^{nx}}\\)<\/li>\n
    6. \\(f_n(x) = 2nxe^{-nx^2}\\)<\/li>\n<\/ol>\n<\/div>\n

      \uc2e4\uc218\uc5f4\uc758 \uc218\ub834\uacfc \uad00\ub828\ud558\uc5ec \ucf54\uc2dc \uc218\uc5f4 \uc870\uac74\uc744 \uc0ac\uc6a9\ud55c \uac83\ucc98\ub7fc \ud568\uc218\uc5f4\uc758 \uade0\ub4f1\uc218\ub834\uc5d0 \ub300\ud574\uc11c\ub3c4 \ucf54\uc2dc \uc870\uac74<\/span>(Cauchy criterion)\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \uc989 \ud568\uc218\uc5f4 \\(\\left\\{ f_n \\right\\}\\)\uc774 \\(I\\)\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740, \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n>N\\), \\(m>N\\)\uc77c \ub54c\ub9c8\ub2e4 \\(\\lVert f_n – f_m \\rVert_\\infty < \\varepsilon\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n

      \uc810\ubcc4\uc218\ub834\uc5d0 \ube44\ud558\uc5ec \uade0\ub4f1\uc218\ub834\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \ub354 \uc88b\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

      \n

      \uc815\ub9ac 8.1. (\uade0\ub4f1\uc218\ub834\uacfc \uc5f0\uc18d\uc131\uc758 \uad00\uacc4)<\/span><\/p>\n

      \ubaa8\ub4e0 \\(f_n\\)\uc774 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\(\\left\\{ f_n \\right\\}\\)\uc774 \\(f\\)\uc5d0 \uade0\ub4f1\uc218\ub834\ud558\uba74 \\(f\\)\ub3c4 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \uc810 \\(c\\in I\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<\/p>\n

      \\(\\varepsilon > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \\(f_n \\rightrightarrows f\\)\uc774\ubbc0\ub85c, \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc77c \ub54c \ubaa8\ub4e0 \\(x \\in I\\)\uc5d0 \ub300\ud574
      \n\\[|f_n(x) – f(x)| < \\frac{\\varepsilon}{3}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(n = N + 1\\)\ub85c \uace0\uc815\uc2dc\ud0a4\uc790. \\(f_{N+1}\\)\uc774 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(|x – c| < \\delta\\)\uc77c \ub54c\n\\[|f_{N+1}(x) - f_{N+1}(c)| < \\frac{\\varepsilon}{3}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(|x - c| < \\delta\\)\uc77c \ub54c\n\\[\\begin{aligned}\n|f(x) - f(c)| &\\leq |f(x) - f_{N+1}(x)| + |f_{N+1}(x) - f_{N+1}(c)| + |f_{N+1}(c) - f(c)|\\\\\n&< \\frac{\\varepsilon}{3} + \\frac{\\varepsilon}{3} + \\frac{\\varepsilon}{3} = \\varepsilon\n\\end{aligned}\\]\n\uc774\ubbc0\ub85c, \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

      \n

      \ubb38\uc81c 8.4.<\/span>
      \n\uad6c\uac04 \\([0,\\,1]\\)\uc5d0\uc11c \ud568\uc218 \\(f_n\\)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.
      \n\\[f_n (x) = \\begin{cases} n &\\,\\,\\text{if }\\,0 < x < \\frac{1}{n} ,\\\\[6pt] 0&\\,\\,\\text{if }\\,x=0\\,\\text{ or }\\,x\\ge\\frac{1}{n}.\\end{cases}\\]\n\uc774\ub54c \\([0,\\,1]\\)\uc5d0\uc11c \\(\\left\\{f_n\\right\\}\\)\uc758 \uadf9\ud55c\ud568\uc218\uac00 \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0\ub9cc \\(\\left\\{f_n\\right\\}\\)\uc758 \uc801\ubd84\uc758 \uadf9\ud55c\uacfc \\(\\left\\{f_n\\right\\}\\)\uc758 \uadf9\ud55c\ud568\uc218\uc758 \uc801\ubd84\uc774 \uc77c\uce58\ud558\uc9c0 \uc54a\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

      \n

      \uc815\ub9ac 8.2. (\uade0\ub4f1\uc218\ub834\uacfc \uc801\ubd84\uc758 \uad00\uacc4)<\/span><\/p>\n

      \ubaa8\ub4e0 \\(f_n\\)\uc774 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \\(\\left\\{ f_n \\right\\}\\)\uc774 \\(f\\)\uc5d0 \uade0\ub4f1\uc218\ub834\ud558\uba74 \\(f\\)\ub3c4 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
      \n\\[\\lim_{n\\rightarrow\\infty}\\int_a^b f_n (x) dx = \\int_a^b f(x)dx .\\tag{8.1}\\]<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \uba3c\uc800 \\(f\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uc790. \\(\\varepsilon > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uade0\ub4f1\uc218\ub834\uc758 \uc815\uc758\uc5d0 \uc758\ud574 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc77c \ub54c \ubaa8\ub4e0 \\(x \\in [a,\\,b]\\)\uc5d0\uc11c
      \n\\[|f_n(x) – f(x)| < \\frac{\\varepsilon}{3(b-a)}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(f_{N+1}\\)\uc774 \uc801\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \ubd84\ud560 \\(P\\)\uac00 \uc874\uc7ac\ud558\uc5ec\n\\[U(f_{N+1},\\, P) - L(f_{N+1},\\, P) < \\frac{\\varepsilon}{3}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc784\uc758\uc758 \uc18c\uad6c\uac04 \\([x_{i-1},\\, x_i]\\)\uc5d0\uc11c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.\n\\[\\begin{aligned}\n\\sup_{[x_{i-1},\\,x_i]} f(x) &\\leq \\sup_{[x_{i-1},\\,x_i]} f_{N+1}(x) + \\frac{\\varepsilon}{3(b-a)}, \\\\[6pt]\n\\inf_{[x_{i-1},\\,x_i]} f(x) &\\geq \\inf_{[x_{i-1},\\,x_i]} f_{N+1}(x) - \\frac{\\varepsilon}{3(b-a)}.\n\\end{aligned}\\]\n\uc774\ub54c \uc0c1\ud569\uacfc \ud558\ud569\uc758 \ucc28\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n\\[\\begin{aligned}\nU(f,\\, P) - L(f,\\, P) &\\leq U(f_{N+1},\\, P) + \\frac{\\varepsilon}{3} - \\left(L(f_{N+1},\\, P) - \\frac{\\varepsilon}{3}\\right)\\\\[6pt]\n&= U(f_{N+1},\\, P) - L(f_{N+1},\\, P) + \\frac{2\\varepsilon}{3}\\\\[6pt]\n&< \\frac{\\varepsilon}{3} + \\frac{2\\varepsilon}{3} = \\varepsilon.\n\\end{aligned}\\]\n\ub530\ub77c\uc11c \\(f\\)\ub294 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n

      \ub2e4\uc74c\uc73c\ub85c \ub4f1\uc2dd (8.1)\uc744 \uc99d\uba85\ud558\uc790. \ub2e4\uc2dc \\(\\varepsilon > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uade0\ub4f1\uc218\ub834\uc758 \uc815\uc758\uc5d0 \uc758\ud574 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc77c \ub54c \ubaa8\ub4e0 \\(x \\in [a,\\,b]\\)\uc5d0\uc11c
      \n\\[|f_n(x) – f(x)| < \\frac{\\varepsilon}{b-a}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc801\ubd84\uc758 \uc131\uc9c8\uc744 \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.\n\\[\\begin{aligned}\n\\left|\\int_a^b f_n(x) dx - \\int_a^b f(x) dx\\right| &= \\left|\\int_a^b (f_n(x) - f(x)) dx\\right|\\\\[6pt]\n&\\leq \\int_a^b |f_n(x) - f(x)| dx\\\\[6pt]\n&< \\int_a^b \\frac{\\varepsilon}{b-a} dx = \\varepsilon .\n\\end{aligned}\\]\n\uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \\(\\lim_{n \\to \\infty} \\int_a^b f_n(x) dx = \\int_a^b f(x) dx\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

      \n

      \uc815\ub9ac 8.3. (\uade0\ub4f1\uc218\ub834\uacfc \ubbf8\ubd84\uc758 \uad00\uacc4)<\/span><\/p>\n

      \ubaa8\ub4e0 \\(f_n\\)\uc774 \\([a,\\,b]\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \uc2e4\uc218\uc5f4 \\(\\left\\{ f_n (x_0 ) \\right\\}\\)\uc774 \uc218\ub834\ud558\ub3c4\ub85d \ud558\ub294 \uc810 \\(x_0 \\in [a,\\,b]\\)\uac00 \uc874\uc7ac\ud558\uba70 \\(\\left\\{ f_n ‘ \\right\\}\\)\uc774 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uade0\ub4f1\uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\left\\{f_n\\right\\}\\)\uc740 \\([a,\\,b]\\)\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud558\uba70 \\(\\left\\{f_n\\right\\}\\)\uc758 \uadf9\ud55c\ud568\uc218 \\(f\\)\ub294 \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \uc784\uc758\uc758 \\(x\\in [a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(f_n ‘ (x) \\rightarrow f ‘ (x)\\)\uc774\ub2e4.<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \\(\\left\\{ f_n ‘ \\right\\}\\)\uc758 \uadf9\ud55c\ud568\uc218\ub97c \\(g\\)\ub77c\uace0 \ub193\uc790. \uc989 \\(f_n’ \\rightrightarrows g\\)\ub77c\uace0 \ud558\uc790.<\/p>\n

      \uba3c\uc800 \\(\\{f_n\\}\\)\uc774 \uade0\ub4f1\uc218\ub834\ud568\uc744 \ubcf4\uc774\uc790.
      \n\\(\\{f_n(x_0)\\}\\)\uc774 \uc218\ub834\ud558\ubbc0\ub85c \ucf54\uc2dc \uc218\uc5f4\uc774\ub2e4. \uc989, \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud574 \\(N_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(m > N_1\\), \\(n > N_1\\)\uc77c \ub54c
      \n\\[|f_m(x_0) – f_n(x_0)| < \\frac{\\varepsilon}{2}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(\\{f_n'\\}\\)\uc774 \uade0\ub4f1\uc218\ub834\ud558\ubbc0\ub85c, \\(N_2\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(m,\\, n > N_2\\)\uc77c \ub54c \ubaa8\ub4e0 \\(t \\in [a,\\,b]\\)\uc5d0\uc11c
      \n\\[|f_m'(t) – f_n'(t)| < \\frac{\\varepsilon}{2(b-a)}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n\\(N = \\max\\{N_1,\\, N_2\\}\\)\ub85c \ub193\uace0 \\(m,\\, n > N\\)\uc774\ub77c \ud558\uc790. \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec, \uc784\uc758\uc758 \\(x \\in [a,\\,b]\\)\uc5d0 \ub300\ud574 \uc801\ub2f9\ud55c \\(c\\)\uac00 \\(x_0\\)\uc640 \\(x\\) \uc0ac\uc774\uc5d0 \uc874\uc7ac\ud558\uc5ec
      \n\\[(f_m – f_n)(x) – (f_m – f_n)(x_0) = (f_m’ – f_n’)(c) \\cdot (x – x_0)\\]
      \n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ub530\ub77c\uc11c
      \n\\[\\begin{aligned}
      \n|f_m(x) – f_n(x)| &\\leq |f_m(x_0) – f_n(x_0)| + |f_m'(c) – f_n'(c)| \\cdot |x – x_0|\\\\[6pt]
      \n&< \\frac{\\varepsilon}{2} + \\frac{\\varepsilon}{2(b-a)} \\cdot (b-a) = \\varepsilon\n\\end{aligned}\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ucf54\uc2dc \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \\(\\{f_n\\}\\)\uc740 \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.<\/p>\n

      \uc774\uc81c \\(f’ = g\\)\uc784\uc744 \ubcf4\uc774\uc790. \uc784\uc758\uc758 \\(x \\in [a,\\,b]\\)\uc640 \ucda9\ubd84\ud788 \uc791\uc740 \\(h \\neq 0\\)\uc5d0 \ub300\ud574
      \n\\[\\begin{align*}
      \n\\frac{f(x+h) – f(x)}{h} – g(x) =& \\left(\\frac{f(x+h) – f(x)}{h} – \\frac{f_n(x+h) – f_n(x)}{h}\\right)\\\\[6pt]
      \n&+ \\left(\\frac{f_n(x+h) – f_n(x)}{h} – f_n'(x)\\right) + (f_n'(x) – g(x))\\tag{8.2}
      \n\\end{align*}\\]
      \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub4f1\uc2dd (8.2)\uc758 \uc6b0\ubcc0\uc758 \uccab \ubc88\uc9f8 \ud56d\uc5d0\uc11c \\(f_n \\rightrightarrows f\\)\uc774\ubbc0\ub85c, \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud574 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
      \n\\[\\left|\\frac{f(x+h) – f(x)}{h} – \\frac{f_n(x+h) – f_n(x)}{h}\\right| < \\frac{\\varepsilon}{3}.\\]\n\ub4f1\uc2dd (8.2)\uc758 \uc6b0\ubcc0\uc758 \uc138 \ubc88\uc9f8 \ud56d\uc5d0\uc11c \\(f_n' \\rightrightarrows g\\)\uc774\ubbc0\ub85c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.\n\\[|f_n'(x) - g(x)| < \\frac{\\varepsilon}{3}.\\tag{8.3}\\]\n\ub4f1\uc2dd (8.2)\uc758 \uc6b0\ubcc0\uc758 \ub450 \ubc88\uc9f8 \ud56d\uc5d0\uc11c \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud574 \uc801\ub2f9\ud55c \\(\\theta \\in (0,\\,1)\\)\uc774 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.\n\\[\\frac{f_n(x+h) - f_n(x)}{h} = f_n'(x + \\theta h).\\tag{8.4}\\]\n\\(f_n'\\)\uc774 \uade0\ub4f1\uc218\ub834\ud558\uace0 \\(g\\)\uac00 \uc5f0\uc18d\uc774\ubbc0\ub85c, \ucda9\ubd84\ud788 \uc791\uc740 \\(|h|\\)\uc5d0 \ub300\ud574 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.\n\\[|f_n'(x + \\theta h) - f_n'(x)| < \\frac{\\varepsilon}{3}.\\tag{8.5}\\]\n(8.2), (8.3), (8.4), (8.5)\ub97c \uacb0\ud569\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.\n\\[\\left|\\frac{f(x+h) - f(x)}{h} - g(x)\\right| < \\varepsilon.\\]\n\uc989 \\(h \\to 0\\)\uc77c \ub54c \\(\\frac{f(x+h) - f(x)}{h} \\to g(x)\\)\uc774\ubbc0\ub85c \\(f\\)\ub294 \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f'(x) = g(x)\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

      \ub9cc\uc57d \ud568\uc218\uc5f4\uc774 \ub2e8\uc870\ub77c\uba74, \ub2e4\uc74c\uacfc \uac19\uc740 \uc720\uc6a9\ud55c \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

      \n

      \uc815\ub9ac 8.4. (\ub514\ub2c8 \uc815\ub9ac)<\/span><\/p>\n

      \ud568\uc218\uc5f4 \\(\\left\\{ f_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \ucef4\ud329\ud2b8 \uacf5\uac04 \\(K\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uc2e4\uc22b\uac12\uc744 \uac16\ub294 \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub610\ud55c \uc784\uc758\uc758 \\(x\\in K\\)\uc640 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(f_n (x) \\le f_{n+1} (x)\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(K\\)\uc5d0\uc11c \\(\\left\\{ f_n \\right\\}\\)\uc774 \uc5f0\uc18d\ud568\uc218 \\(f\\)\ub85c \uc810\ubcc4\uc218\ub834\ud558\uba74, \\(K\\)\uc5d0\uc11c \\(\\left\\{ f_n \\right\\}\\)\uc740 \\(f\\)\ub85c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \\(g_n(x) = f(x) – f_n(x)\\)\ub77c \ud558\uc790.
      \n\uac01 \\(g_n\\)\uc740 \uc5f0\uc18d\ud568\uc218\uc758 \ucc28\uc774\ubbc0\ub85c \uc5f0\uc18d\uc774\ub2e4.
      \n\ub610\ud55c \\(f_n \\leq f_{n+1}\\)\uc774\ubbc0\ub85c \\(g_{n+1}(x) \\leq g_n(x)\\). \uc989, \\(\\{g_n\\}\\)\uc740 \uac10\uc18c\uc218\uc5f4\uc774\ub2e4.
      \n\uadf8\ub9ac\uace0 \uc810\ubcc4\uc218\ub834 \uc870\uac74\uc5d0 \uc758\ud574 \uac01 \\(x \\in K\\)\uc5d0\uc11c \\(g_n(x) \\to 0\\)\uc774\ub2e4.<\/p>\n

      \\(\\varepsilon > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uac01 \\(x \\in K\\)\uc5d0 \ub300\ud574 \uc810\ubcc4\uc218\ub834 \uc870\uac74\uc5d0 \uc758\ud574 \uc790\uc5f0\uc218 \\(N_x\\)\uac00 \uc874\uc7ac\ud558\uc5ec
      \n\\[g_{N_x}(x) = f(x) – f_{N_x}(x) < \\frac{\\varepsilon}{2}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(g_{N_x}\\)\uac00 \\(x\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c, \\(x\\)\uc758 \uc5f4\ub9b0\uadfc\ubc29 \\(U_x\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \\(y \\in U_x\\)\uc5d0 \ub300\ud574\n\\[g_{N_x}(y) < \\frac{3\\varepsilon}{4}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc5ec\uae30\uc11c \\(\\{U_x \\mid x \\in K\\}\\)\ub294 \\(K\\)\uc758 \uc5f4\ub9b0\ub36e\uac1c\uc774\ub2e4. \\(K\\)\uac00 \ucef4\ud329\ud2b8\uc774\ubbc0\ub85c \uc774 \uc5f4\ub9b0\ub36e\uac1c\uc758 \uc720\ud55c\ubd80\ubd84\ub36e\uac1c\uac00 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.\n\\[K \\subseteq U_{x_1} \\cup U_{x_2} \\cup \\cdots \\cup U_{x_m}.\\]\n\\(N = \\max\\{N_{x_1},\\, N_{x_2},\\, \\ldots,\\, N_{x_m}\\}\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

      \\(y \\in K\\)\uc774\uace0 \\(n > N\\)\uc77c \ub54c, \\(y \\in U_{x_i}\\)\uc778 \uc5b4\ub5a4 \\(i\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.
      \n\\(\\{g_n\\}\\)\uc774 \uac10\uc18c\uc218\uc5f4\uc774\uace0 \\(n > N \\geq N_{x_i}\\)\uc774\ubbc0\ub85c
      \n\\[g_n(y) \\leq g_N(y) \\leq g_{N_{x_i}}(y) < \\frac{3\\varepsilon}{4} < \\varepsilon\\]\n\uc774\ub2e4. \ub610\ud55c \\(g_n(y) = f(y) - f_n(y) \\geq 0\\)\uc774\ub2e4.\n\ub530\ub77c\uc11c \\(n > N\\)\uc77c \ub54c \ubaa8\ub4e0 \\(y \\in K\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[|f_n(y) – f(y)| = g_n(y) < \\varepsilon\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\{f_n\\}\\)\uc740 \\(K\\)\uc5d0\uc11c \\(f\\)\ub85c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.<\/span><\/p>\n<\/div>\n

      \uac70\ub4ed\uc81c\uacf1\uae09\uc218<\/h3>\n

      \ud568\uc218\uc5f4 \\(\\{f_n\\}\\)\uc758 \uae09\uc218 \\(\\sum_{n=1}^{\\infty} f_n\\)\uc744 \ud568\uc218\uae09\uc218<\/span>(series of functions)\ub77c\uace0 \ubd80\ub978\ub2e4. \ud568\uc218\uae09\uc218\uc758 \uc810\ubcc4\uc218\ub834\uacfc \uade0\ub4f1\uc218\ub834\ub3c4 \ud568\uc218\uc5f4\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc815\uc758\ud55c\ub2e4. \ub610\ud55c \ud568\uc218\uc5f4\uc758 \uade0\ub4f1\uc218\ub834\uc5d0 \ub300\ud574\uc11c\ub3c4 \ucf54\uc2dc \uc870\uac74\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \ub354\uc6b1\uc774, \ud568\uc218\uc5f4\uc758 \uade0\ub4f1\uc218\ub834\uacfc \uad00\ub828\ud574\uc11c\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc720\uc6a9\ud55c \ud310\uc815\ubc95\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n

      \n

      \uc815\ub9ac 8.5. (\ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4 \\(M\\)-\ud310\uc815\ubc95)<\/span><\/p>\n

      \\(\\left\\{ f_n \\right\\}\\)\uc774 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ud568\uc218\uc5f4\uc774\uace0, \\(\\left\\{ M_n \\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \\(n\\)\uacfc \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(|f_n(x)| \\leq M_n\\)\uc774\uace0 \\(\\sum M_n\\)\uc774 \uc218\ub834\ud558\uba74 \\(\\sum f_n\\)\uc740 \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \ubd80\ub4f1\uc2dd
      \n\\[\\left\\lVert \\sum_{k=m}^{n} f_k \\right\\rVert \\le \\sum_{k=m}^{n} \\left\\lVert f_k \\right\\rVert \\le \\sum_{k=m}^n M_k\\]
      \n\ub97c \uc0ac\uc6a9\ud558\uba74 \\(\\sum f_n\\)\uc758 \ubd80\ubd84\ud569\uc774 \ucf54\uc2dc \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0b4\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.<\/span><\/p>\n<\/div>\n

      \uc218\uc5f4 \\(\\left\\{a_n\\right\\}\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[\\sum_{n=0}^{\\infty} a_n (x – c)^n\\tag{8.6}\\]
      \n\uaf34\uc758 \ubb34\ud55c\uae09\uc218\ub97c \uac70\ub4ed\uc81c\uacf1\uae09\uc218<\/span>(power series) \ub610\ub294 \uba71\uae09\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \uc5ec\uae30\uc11c \\(c\\)\ub97c \uc774 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc911\uc2ec<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

      \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub294 \\(x\\)\uc758 \uac12\uc5d0 \ub530\ub77c \uc218\ub834\ud560 \uc218\ub3c4 \uc788\uace0 \ubc1c\uc0b0\ud560 \uc218\ub3c4 \uc788\ub2e4. \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uac00 \uc218\ub834\ud558\ub3c4\ub85d \ud558\ub294 \\(x\\)\uc758 \uc9d1\ud569\uc740 \ubc18\ub4dc\uc2dc \uad6c\uac04\uc758 \ud615\ud0dc\uac00 \ub41c\ub2e4. \uc774 \uad6c\uac04\uc744 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc218\ub834\uad6c\uac04<\/span>(interval of convergence)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

      \uc911\uc2ec\uc774 \\(c\\)\uc778 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \uc911 \ud558\ub098\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

        \n
      • \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uac00 \uc624\uc9c1 \\(x = c\\)\uc5d0\uc11c\ub9cc \uc218\ub834\ud55c\ub2e4.<\/li>\n
      • \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uac00 \uc218\ub834\ud55c\ub2e4.<\/li>\n
      • \uc801\ub2f9\ud55c \\(R > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(|x-c|R\\)\uc77c \ub54c \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uac00 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n<\/ul>\n

        \uc138 \ubc88\uc9f8 \uacbd\uc6b0\uc5d0\uc11c \\(R\\)\uc744 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc218\ub834\ubc18\uc9c0\ub984<\/span>(radius of convergence)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uccab \ubc88\uc9f8 \uacbd\uc6b0\uc5d0\uc11c\ub294 \uc218\ub834\ubc18\uc9c0\ub984\uc744 \\(0\\)\uc73c\ub85c \uc815\uc758\ud558\uace0, \ub450 \ubc88\uc9f8 \uacbd\uc6b0\uc5d0\uc11c\ub294 \uc218\ub834\ubc18\uc9c0\ub984\uc744 \\(\\infty\\)\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

        \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc218\ub834\ubc18\uc9c0\ub984\uc774 \uc591\uc218 \\(R\\)\uc77c \ub54c, \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc218\ub834\uad6c\uac04\uc740 \ub124 \uac00\uc9c0 \ud615\ud0dc \uc911 \ud558\ub098\uc774\ub2e4.
        \n\\[ [c-R,\\, c+R] ,\\,\\, (c-R,\\, c+R),\\,\\, [c-R ,\\,c+R) ,\\,\\, (c-R ,\\, c+R ].\\]
        \n\uadf8\ub7ec\ubbc0\ub85c \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc218\ub834\uad6c\uac04\uc744 \uad6c\ud560 \ub54c\ub294 \uc218\ub834\ubc18\uc9c0\ub984\uc744 \uad6c\ud55c \ub4a4, \uc218\ub834\ubc18\uc9c0\ub984\uc774 \uc591\uc758 \uc2e4\uc218\uc774\uba74 \uc218\ub834\uad6c\uac04\uc758 \uc591\ucabd \ub05d\uc810\uc5d0\uc11c\uc758 \uc218\ub834\uc131\ub9cc \ud655\uc778\ud558\uba74 \ucda9\ubd84\ud558\ub2e4.<\/p>\n

        \uac70\ub4ed\uc81c\uacf1\uae09\uc218 \\(\\sum a_n (x-c)^n\\)\uc758 \uc218\ub834\ubc18\uc9c0\ub984\uc744 \uad6c\ud558\ub294 \ubc29\ubc95\uc73c\ub85c \uc720\uc6a9\ud55c \uacf5\uc2dd\uc774 \uc788\ub2e4. \uc989
        \n\\[\\rho = \\varlimsup_{n\\rightarrow\\infty} \\sqrt[n]{\\left|a_n\\right|}\\]
        \n\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

          \n
        • \ub9cc\uc57d \\(0<\\rho<\\infty\\)\uc774\uba74 \uc218\ub834\ubc18\uc9c0\ub984\uc740 \\(R = \\frac{1}{\\rho}\\)\uc774\ub2e4.<\/li>\n
        • \ub9cc\uc57d \\(\\rho =0\\)\uc774\uba74 \uc218\ub834\ubc18\uc9c0\ub984\uc740 \\(\\infty\\)\uc774\ub2e4.<\/li>\n
        • \ub9cc\uc57d \\(\\rho=\\infty\\)\uc774\uba74 \uc218\ub834\ubc18\uc9c0\ub984\uc740 \\(0\\)\uc774\ub2e4.<\/li>\n<\/ul>\n

          \uc774 \uacb0\uacfc\ub294 \ubb34\ud55c\uae09\uc218\uc758 \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc744 \uc0ac\uc6a9\ud558\uba74 \uc27d\uac8c \uc99d\uba85\ub41c\ub2e4.
          \n\uc774 \uacf5\uc2dd\uc744 \ucf54\uc2dc-\uc544\ub2e4\ub9c8\ub974 \uacf5\uc2dd<\/span>(Cauchy-Hadamard formula)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

          \n

          \ubb38\uc81c 8.5.<\/span>
          \n\ub2e4\uc74c \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc218\ub834\uad6c\uac04\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n

            \n
          1. \\(\\sum_{n=1}^{\\infty} \\frac{x^n}{n!}\\)<\/li>\n
          2. \\(\\sum_{n=1}^{\\infty} \\frac{x^n}{\\sqrt{n}}\\)<\/li>\n
          3. \\(\\sum_{n=1}^{\\infty} n!\\,x^n\\)<\/li>\n
          4. \\(\\sum_{n=1}^{\\infty} \\frac{x^n}{2^n}\\)<\/li>\n
          5. \\(\\sum_{n=1}^{\\infty} \\frac{x^n}{n2^n}\\)<\/li>\n
          6. \\(\\sum_{n=1}^{\\infty} \\frac{3^n\\, x^n}{n4^n}\\)<\/li>\n
          7. \\(\\sum_{n=1}^{\\infty} \\frac{x^n}{n^2 +1}\\)<\/li>\n
          8. \\(\\sum_{n=1}^{\\infty} \\frac{x^n}{\\ln(n+1)}\\)<\/li>\n
          9. \\(\\sum_{n=1}^{\\infty} \\frac{(x+3)^{2n}}{5^n}\\)<\/li>\n<\/ol>\n<\/div>\n

            \ub2e4\uc74c \uc815\ub9ac\ub294 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uade0\ub4f1\uc218\ub834\uacfc \uad00\ub828\ub41c \ub9e4\uc6b0 \uc720\uc6a9\ud55c \uacb0\uacfc\ub97c \uc81c\uacf5\ud55c\ub2e4.<\/p>\n

            \n

            \uc815\ub9ac 8.6. (\uc544\ubca8\uc758 \uc815\ub9ac)<\/span><\/p>\n

            \\(R>0\\)\uc774\uace0 \uac70\ub4ed\uc81c\uacf1\uae09\uc218 \\(\\sum a_n (x-c)^n\\)\uc774 \\(x = c+R\\)\uc5d0\uc11c \uc218\ub834\ud558\uba74, \uc774 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub294 \\([c,\\, c+R]\\)\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

            \n

            \uc99d\uba85<\/p>\n

            \ud3b8\uc758\uc0c1 \\(c = 0\\)\uc73c\ub85c \ub193\uc790.
            \n\ubd80\ubd84\ud569\uc744 \\(S_N(x) = \\sum_{n=0}^{N} a_n x^n\\)\uc774\ub77c\uace0 \ud558\uace0, \\(B_N = \\sum_{n=0}^{N} a_n R^n\\)\uc774\ub77c\uace0 \ud558\uc790. \uac00\uc815\uc5d0 \uc758\ud574 \\(B_N \\to B\\)\uc778 \\(B\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.
            \n\\(0 \\leq x \\leq R\\)\uc778 \\(x\\)\uc5d0 \ub300\ud574 \ub2e4\uc74c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.
            \n\\[\\begin{aligned}
            \nS_N(x) &= \\sum_{n=0}^{N} a_n x^n = \\sum_{n=0}^{N} a_n R^n \\cdot \\left(\\frac{x}{R}\\right)^n\\\\[6pt]
            \n&= \\sum_{n=0}^{N} (B_n – B_{n-1}) \\cdot \\left(\\frac{x}{R}\\right)^n .\\quad (\\text{\ub2e8, }B_{-1} = 0.)
            \n\\end{aligned}\\]
            \n\uc544\ubca8\uc758 \ubd80\ubd84\ud569 \uacf5\uc2dd\uc744 \uc0ac\uc6a9\ud558\uba74 \uc704 \uc2dd\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ubcc0\ud615\ud560 \uc218 \uc788\ub2e4.
            \n\\[\\begin{aligned}
            \nS_N(x) &= B_N \\left(\\frac{x}{R}\\right)^N + \\sum_{n=0}^{N-1} B_n \\left[\\left(\\frac{x}{R}\\right)^n – \\left(\\frac{x}{R}\\right)^{n+1}\\right]\\\\[6pt]
            \n&= B_N \\left(\\frac{x}{R}\\right)^N + \\sum_{n=0}^{N-1} B_n \\left(\\frac{x}{R}\\right)^n \\left(1 – \\frac{x}{R}\\right).
            \n\\end{aligned}\\]
            \n\\(B_N \\to B\\)\uc774\ubbc0\ub85c, \\(0 \\leq x < R\\)\uc778 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(N \\to \\infty\\)\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.\n\\[B_N \\left(\\frac{x}{R}\\right)^N \\to 0.\\]\n\ub530\ub77c\uc11c \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.\n\\[S(x) = \\sum_{n=0}^{\\infty} a_n x^n = \\left(1 - \\frac{x}{R}\\right) \\sum_{n=0}^{\\infty} B_n \\left(\\frac{x}{R}\\right)^n .\\]\n\\(\\varepsilon > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \\(N\\rightarrow\\infty\\)\uc77c \ub54c \\(B_N\\)\uc774 \uc218\ub834\ud558\ubbc0\ub85c, \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(|B_n| \\leq K\\)\uc778 \\(K > 0\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \ub610\ud55c \\(N_0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(n > N_0\\)\uc77c \ub54c \\(|B_n – B| < \\varepsilon\\)\uc774\ub2e4.\n\\(N > M > N_0\\)\uc774\uace0 \\(x \\in [0,\\, R]\\)\uc77c \ub54c \ub2e4\uc74c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.
            \n\\[\\begin{aligned}
            \n|S_N(x) – S_M(x)| &= \\left|\\sum_{n=M+1}^{N} a_n x^n\\right|\\\\[6pt]
            \n&= \\left|B_N \\left(\\frac{x}{R}\\right)^N – B_M \\left(\\frac{x}{R}\\right)^M + \\sum_{n=M+1}^{N-1} B_n \\left(\\frac{x}{R}\\right)^n \\left(1 – \\frac{x}{R}\\right)\\right|
            \n\\end{aligned}\\]
            \n\\(x = R\\)\uc77c \ub54c \\(|S_N(R) – S_M(R)| = |B_N – B_M| < \\varepsilon\\)\uc774\ub2e4.<\/p>\n

            \\(0 \\leq x < R\\)\uc77c \ub54c \\(t = \\frac{x}{R} < 1\\)\ub85c \ub193\uc73c\uba74\n\\[\\begin{aligned}\n|S_N(x) - S_M(x)| &\\leq |B_N| t^N + |B_M| t^M + (1-t) \\sum_{n=M+1}^{N-1} |B_n| t^n\\\\[6pt]\n&\\leq 2K t^M + (1-t) \\cdot 2K \\cdot \\frac{t^{M+1}}{1-t}\\\\[6pt]\n&= 2K t^M (1 + t) \\leq 4K t^M\n\\end{aligned}\\]\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(M\\)\uc744 \ucda9\ubd84\ud788 \ud06c\uac8c \uc120\ud0dd\ud558\uba74 \ubaa8\ub4e0 \\(t \\in (0,\\, 1)\\)\uc5d0 \ub300\ud574 \\(4K t^M < \\varepsilon\\)\uc774\ub2e4.<\/p>\n

            \uadf8\ub7ec\ubbc0\ub85c \ucf54\uc2dc \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \\(\\sum a_n x^n\\)\uc740 \\([0,\\, R]\\)\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.<\/span><\/p>\n<\/div>\n

            \uade0\ub4f1\uc218\ub834\uc758 \uc131\uc9c8\uacfc \uc544\ubca8\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

            \n

            \uc815\ub9ac 8.7. (\uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uae30\ubcf8\uc131\uc9c8)<\/span><\/p>\n

            \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub294 \ubbf8\uc801\ubd84\uacfc \uad00\ub828\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

              \n
            1. \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub294 \uc218\ub834\uad6c\uac04\uc5d0 \ud3ec\ud568\ub418\ub294 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.<\/li>\n
            2. \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub294 \uc218\ub834\uad6c\uac04\uc758 \ub0b4\ubd80\uc5d0\uc11c \ud56d\ubcc4\ub85c \ubbf8\ubd84\ud558\uac70\ub098 \uc801\ubd84\ud560 \uc218 \uc788\ub2e4.<\/li>\n
            3. \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \ud56d\ubcc4\ub85c \ubbf8\ubd84\ud574\ub3c4 \uc218\ub834\ubc18\uc9c0\ub984\uc774 \ubcc0\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
              \n

              \ubb38\uc81c 8.6.<\/span>
              \n\uac70\ub4ed\uc81c\uacf1\uae09\uc218 \\(\\sum_{n=0}^{\\infty} a_n (x-c)^n\\)\uc758 \uc218\ub834\uad6c\uac04\uc774 \\(I\\)\uc774\uace0, \\(K\\)\uac00 \\(I\\)\uc758 \ucef4\ud329\ud2b8 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\sum_{n=0}^{\\infty} a_k (x-c)^n\\)\uc774 \\(K\\)\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

              \n

              \ubb38\uc81c 8.7.<\/span>
              \n\uc2e4\uc218\uc5f4 \\(\\left\\{a_n\\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \\(\\displaystyle\\varlimsup_{n\\rightarrow\\infty}\\left( n\\lvert a_n \\rvert \\right)^{\\frac{1}{n}} = \\varlimsup_{n\\rightarrow\\infty} \\lvert a_n \\rvert^{\\frac{1}{n}}\\)\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

              \n

              \ubb38\uc81c 8.8.<\/span>
              \n\uac70\ub4ed\uc81c\uacf1\uae09\uc218 \\(\\sum_{n=0}^{\\infty} a_n (x-c)^n\\)\uc758 \uc218\ub834\ubc18\uc9c0\ub984 \\(R\\)\uc774 \uc591\uc758 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \uc5f4\ub9b0\uad6c\uac04 \\((c-R ,\\,c+R)\\)\uc5d0\uc11c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.
              \n\\[ \\frac{d}{dx}\\sum_{n=0}^{\\infty} a_n (x-c)^n = \\sum_{n=1}^{\\infty} na_n (x-c)^{n-1}.\\]
              \n\ub610\ud55c \uc704 \ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc758 \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834\ubc18\uc9c0\ub984\ub3c4 \\(R\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

              \n

              \ubb38\uc81c 8.9.<\/span>
              \n\uac70\ub4ed\uc81c\uacf1\uae09\uc218 \\(\\sum_{n=0}^{\\infty} a_n (x-c)^n\\)\uc758 \uc218\ub834\uad6c\uac04\uc774 \\(I\\)\uc774\uace0 \\(a\\)\uc640 \\(b\\)\uac00 \\(I\\)\uc758 \uc810\uc774\uba70 \\(a\n<\/div>\n

              \ud14c\uc77c\ub7ec \uae09\uc218\uc640 \ud574\uc11d\uc801 \ud568\uc218<\/h3>\n

              \ud568\uc218 \\(f\\)\uac00 \uc810 \\(c\\)\uc5d0\uc11c \\(n\\)\ubc88 \uc774\uc0c1 \ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c, \uc810 \\(c\\)\uc5d0\uc11c \ud568\uc218 \\(f\\)\uc758 \\(n\\)\ucc28 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd<\/span>(Taylor polynomial)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
              \n\\[P_n (x) = \\sum_{k=0}^{n} \\frac{f^{(k)}(c)}{k!} (x-c)^k .\\]
              \n\ub9cc\uc57d \\(f\\)\uac00 \uc810 \\(c\\)\uc5d0\uc11c \uc784\uc758 \ud69f\uc218\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74, \uc704 \uc2dd\uc5d0 \\(n\\rightarrow\\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud55c \ubb34\ud55c\uae09\uc218\ub97c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.
              \n\\[\\sum_{n=0}^{\\infty} \\frac{f^{(n)}(c)}{n!}(x – c)^n.\\]
              \n\uc774 \ubb34\ud55c\uae09\uc218\ub97c \\(c\\)\ub97c \uc911\uc2ec\uc73c\ub85c \ud55c \\(f\\)\uc758 \ud14c\uc77c\ub7ec \uae09\uc218<\/span>(Taylor series)\ub77c\uace0 \ubd80\ub978\ub2e4. \ud2b9\ud788 \\(c=0\\)\uc77c \ub54c, \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \ub9e5\ud074\ub77c\ub9b0 \uae09\uc218<\/span>(Maclaurin series)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

              \uc810 \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \uad6c\uac04 \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(c\\)\ub97c \uc911\uc2ec\uc73c\ub85c \ud558\ub294 \\(f\\)\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \\(f\\)\uc5d0 \uc218\ub834\ud558\uba74, \\(I\\)\uc5d0\uc11c \ud568\uc218 \\(f\\)\ub97c \ud14c\uc77c\ub7ec \uae09\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc774\uac83\uc740 \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ud14c\uc77c\ub7ec \ub098\uba38\uc9c0(\uc815\ub9ac 5.7)\uac00 \\(0\\)\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uacfc \uac19\ub2e4.<\/p>\n

              \uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \ud14c\uc77c\ub7ec \uae09\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
              \n\\[\\begin{aligned}
              \ne^x &= \\sum_{n=0}^{\\infty} \\frac{x^n}{n!} &&(x\\in\\mathbb{R}) \\\\[6pt]
              \n\\sin x &= \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n+1}}{(2n+1)!} &&(x\\in\\mathbb{R}) \\\\[6pt]
              \n\\cos x &= \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n}}{(2n)!} &&(x\\in\\mathbb{R}) \\\\[6pt]
              \n\\tan^{-1} x &= \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n+1}}{2n+1} &&(|x| \\le 1) \\\\[6pt]
              \n\\ln(1 + x) &= \\sum_{n=1}^{\\infty} \\frac{(-1)^{n-1} x^n}{n} &&(-1 < x \\le 1) \\\\[6pt]\n\\frac{1}{1 - x} &= \\sum_{n=0}^{\\infty} x^n &&(|x| < 1) \\\\[6pt]\n(1 + x)^\\alpha &= \\sum_{n=0}^{\\infty} \\binom{\\alpha}{n} x^n &&(|x| < 1)\n\\end{aligned}\\]<\/p>\n

              \ud568\uc218 \\(f\\)\uac00 \uc810 \\(c\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \uc810 \\(c\\)\ub97c \uc911\uc2ec\uc73c\ub85c \ud558\ub294 \uc790\uc2e0\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uc640 \uc77c\uce58\ud560 \ub54c, “\\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \ud574\uc11d\uc801<\/span>\uc774\ub2e4(analytic)” \ub610\ub294 “\\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uc2e4\ud574\uc11d\uc801<\/span>\uc774\ub2e4(real analytic)”\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ub9cc\uc57d \ud568\uc218 \\(f\\)\uac00 \uc601\uc5ed \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ud574\uc11d\uc801\uc774\uba74 “\\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ud574\uc11d\uc801\uc774\ub2e4”\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc815\uc758\uc5ed\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ud574\uc11d\uc801\uc778 \ud568\uc218\ub97c \ud574\uc11d\uc801\uc778 \ud568\uc218<\/span>(analytic function)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

              \uc784\uc758 \ud69f\uc218\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uac00 \ud56d\uc0c1 \ud574\uc11d\uc801\uc778 \uac83\uc740 \uc544\ub2c8\ub2e4. \uc608\ub97c \ub4e4\uc5b4,
              \n\\[f(x) = \\begin{cases} e^{-\\frac{1}{x^2}} & \\text{if }\\,x \\neq 0, \\\\[6pt] 0 & \\text{if }\\,x = 0 \\end{cases}\\]
              \n\uc774\ub77c\uace0 \uc815\uc758\ub41c \ud568\uc218 \\(f:\\mathbb{R}\\rightarrow\\mathbb{R}\\)\uc740 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \uc784\uc758 \ud69f\uc218\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4. \ud558\uc9c0\ub9cc \\(c=0\\)\uc77c \ub54c \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(f^{(n)}(c)=0\\)\uc774\ubbc0\ub85c \\(0\\)\uc5d0\uc11c \\(f\\)\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\ub294 \uc0c1\uc218\ud568\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \\(0\\)\uc744 \ud3ec\ud568\ud558\ub294 \uc784\uc758\uc758 \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc790\uc2e0\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uc640 \uc77c\uce58\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc989 \\(f\\)\ub294 \uc784\uc758 \ud69f\uc218\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud558\uc9c0\ub9cc \\(0\\)\uc5d0\uc11c \ud574\uc11d\uc801\uc774\uc9c0 \uc54a\ub2e4.<\/p>\n

              \ud574\uc11d\uc801\uc778 \ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n