{"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":"2026-09-27T15:02:47","modified_gmt":"2026-09-27T06:02:47","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 \uc9d1\ud569 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc2e4\ud568\uc218\uc758 \ud568\uc218\uc5f4\uc774\uace0 \\(f\\colon I\\to\\mathbb R\\)\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[
\n\\lim_{n\\to\\infty}f_n(x)=f(x)
\n\\]
\n\uc774\uba74 \\(I\\)\uc5d0\uc11c \\(\\{f_n\\}\\)\uc774 \\(f\\)\uc5d0 \uc810\ubcc4\uc218\ub834<\/span>\ud55c\ub2e4(converges pointwise)\uace0 \ud55c\ub2e4. \uc774\ub54c \\(f\\)\ub97c \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\ub974\uace0 \\(f_n\\to 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 \\(\\{f_n\\}\\)\uc758 \uadf9\ud55c\ud568\uc218\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

\n

\ubb38\uc81c 8.2.<\/span>
\n\uc218\uc5f4 \\(\\{x_n\\}\\)\uc774 \ubaa8\ub4e0 \uc720\ub9ac\uc218\ub97c \ud55c \ubc88\uc529 \ucde8\ud558\ub294 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \ud568\uc218 \\(f_n\\colon\\mathbb R\\to\\mathbb R\\)\ub97c
\n\\[
\nf_n(x)=
\n\\begin{cases}
\n1 & \\text{if }\\;x\\in\\{x_1,x_2,\\ldots,x_n\\},\\\\[5pt]
\n0 & \\text{if }\\;x\\notin\\{x_1,x_2,\\ldots,x_n\\}
\n\\end{cases}
\n\\]
\n\uc774\ub77c\uace0 \uc815\uc758\ud560 \ub54c \\(\\{f_n\\}\\)\uc758 \uadf9\ud55c\ud568\uc218\ub97c \uad6c\ud558\uc2dc\uc624. \ub610\ud55c \uac01 \\(f_n\\)\uc740 \ubaa8\ub4e0 \ub2eb\ud78c \uc720\uacc4\uad6c\uac04\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0\ub9cc \uadf9\ud55c\ud568\uc218\ub294 \uadf8\ub807\uc9c0 \uc54a\uc74c\uc744 \ud655\uc778\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

\uc810\ubcc4\uc218\ub834\ub9cc\uc73c\ub85c\ub294 \uc5f0\uc18d\uc131, \uc801\ubd84, \ubbf8\ubd84\uacfc \uadf9\ud55c\uc758 \uc21c\uc11c\ub97c \ubc14\uafc0 \uc218 \uc5c6\ub2e4. \uc2e4\uc81c\ub85c \ubaa8\ub4e0 \\(f_n\\)\uc774 \uc5f0\uc18d\uc774\uc5b4\ub3c4 \uadf9\ud55c\ud568\uc218\ub294 \uc5f0\uc18d\uc774 \uc544\ub2d0 \uc218 \uc788\uace0, \ubaa8\ub4e0 \\(f_n\\)\uc774 \uc801\ubd84 \uac00\ub2a5\ud574\ub3c4 \uadf9\ud55c\ud568\uc218\ub294 \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\uc744 \uc218 \uc788\uc73c\uba70, \ubaa8\ub4e0 \\(f_n\\)\uc774 \ubbf8\ubd84 \uac00\ub2a5\ud574\ub3c4 \uadf9\ud55c\ud568\uc218\ub294 \ubbf8\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\uc744 \uc218 \uc788\ub2e4. \uc774\ub7ec\ud55c \ubb38\uc81c\ub97c \ubcf4\uc644\ud558\uae30 \uc704\ud574 \uade0\ub4f1\uc218\ub834\uc744 \uc815\uc758\ud55c\ub2e4.<\/p>\n

\ud568\uc218\uc5f4 \\(\\{f_n\\}\\)\uc774 \\(I\\)\uc5d0\uc11c \ud568\uc218 \\(f\\)\uc5d0 \uade0\ub4f1\uc218\ub834<\/span>\ud55c\ub2e4(converges uniformly)\ub294 \uac83\uc740 \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n>N\\)\uc774\uace0 \\(x\\in I\\)\uc774\uba74
\n\\[
\n|f_n(x)-f(x)|<\\varepsilon
\n\\]
\n\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. \uc774\uac83\uc744 \\(f_n\\rightrightarrows f\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc815\uc758\uc5d0\uc11c \ubc14\ub85c \uc54c \uc218 \uc788\ub4ef\uc774 \uade0\ub4f1\uc218\ub834\ud558\uba74 \uc810\ubcc4\uc218\ub834\ud55c\ub2e4.<\/p>\n

\\(I\\)\uc5d0\uc11c \uc720\uacc4\uc778 \ud568\uc218 \\(g\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[
\n\\|g\\|_\\infty=\\sup_{x\\in I}|g(x)|
\n\\]
\n\ub77c\uace0 \uc815\uc758\ud558\uba74, \\(f_n-f\\)\uac00 \uc720\uacc4\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(f_n\\rightrightarrows f\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc774 \\(\\|f_n-f\\|_\\infty\\to0\\)\uc778 \uac83\uc774\ub2e4.<\/p>\n

\n

\ubb38\uc81c 8.3.<\/span>
\n\uad6c\uac04 \\((0,1]\\)\uc5d0\uc11c \ub2e4\uc74c \ud568\uc218\uc5f4\uc758 \uc810\ubcc4\uadf9\ud55c\uc744 \uad6c\ud558\uace0 \uade0\ub4f1\uc218\ub834 \uc5ec\ubd80\ub97c \ud310\uc815\ud558\uc2dc\uc624.<\/p>\n

    \n
  1. \\(f_n(x)=\\dfrac{1}{n+x}\\)<\/li>\n
  2. \\(f_n(x)=\\dfrac{\\sin nx}{n}\\)<\/li>\n
  3. \\(f_n(x)=nx(1-x)^n\\)<\/li>\n
  4. \\(f_n(x)=\\dfrac{1}{nx+1}\\)<\/li>\n
  5. \\(f_n(x)=\\dfrac1x+\\dfrac{nx}{e^{nx}}\\)<\/li>\n
  6. \\(f_n(x)=2nxe^{-nx^2}\\)<\/li>\n<\/ol>\n<\/div>\n

    \uc2e4\uc218\uc5f4\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c \uade0\ub4f1\uc218\ub834\ub3c4 \ucf54\uc2dc \uc870\uac74\uc73c\ub85c \ud310\uc815\ud560 \uc218 \uc788\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 8.1. (\uade0\ub4f1\uc218\ub834\uc758 \ucf54\uc2dc \ud310\uc815\ubc95)<\/span><\/p>\n

    \uc2e4\ud568\uc218\uc758 \ud568\uc218\uc5f4 \\(\\{f_n\\}\\)\uc774 \\(I\\)\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud560 \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 \\(m,n>N\\)\uc774\uace0 \\(x\\in I\\)\uc774\uba74
    \n\\[
    \n|f_n(x)-f_m(x)|<\\varepsilon
    \n\\]
    \n\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/span>
    \n\\(f_n\\rightrightarrows f\\)\uc774\uba74 \ucda9\ubd84\ud788 \ud070 \\(m,n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n|f_n(x)-f_m(x)|
    \n\\le |f_n(x)-f(x)|+|f_m(x)-f(x)|
    \n<\\varepsilon
    \n\\]
    \n\uc774 \ubaa8\ub4e0 \\(x\\in I\\)\uc5d0\uc11c \uc131\ub9bd\ud558\ubbc0\ub85c \ucf54\uc2dc \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n

    \uc5ed\uc73c\ub85c \uade0\ub4f1 \ucf54\uc2dc \uc870\uac74\uc744 \uac00\uc815\ud558\uc790. \uac01 \\(x\\in I\\)\ub97c \uace0\uc815\ud558\uba74 \\(\\{f_n(x)\\}\\)\uc740 \uc2e4\uc218\uc758 \ucf54\uc2dc \uc218\uc5f4\uc774\ubbc0\ub85c \uc2e4\uc218\uacc4\uc758 \uc644\ube44\uc131\uc5d0 \uc758\ud574 \uc5b4\ub5a4 \\(f(x)\\in\\mathbb R\\)\ub85c \uc218\ub834\ud55c\ub2e4. \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \\(m,n>N\\)\uc774\uba74 \ubaa8\ub4e0 \\(x\\in I\\)\uc5d0\uc11c \\(|f_n(x)-f_m(x)|<\\varepsilon\/2\\)\uac00 \ub418\ub3c4\ub85d \\(N\\)\uc744 \ud0dd\ud558\uc790. \\(m\\to\\infty\\)\ub85c \ubcf4\ub0b4\uba74
    \n\\[
    \n|f_n(x)-f(x)|\\le\\frac{\\varepsilon}{2}<\\varepsilon
    \n\\]
    \n\uc774 \ubaa8\ub4e0 \\(x\\in I\\)\uc640 \\(n>N\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \\(f_n\\rightrightarrows f\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\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 8.2. (\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 \\(f_n\\rightrightarrows f\\)\uc774\uba74 \\(f\\)\ub3c4 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/span>
    \n\uc810 \\(c\\in I\\)\uc640 \\(\\varepsilon>0\\)\uc744 \ud0dd\ud558\uc790. \uade0\ub4f1\uc218\ub834\uc5d0 \uc758\ud574 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n|f_{N+1}(x)-f(x)|<\\frac{\\varepsilon}{3}
    \n\\]
    \n\uc774\ub2e4. \\(f_{N+1}\\)\uc774 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \uc5b4\ub5a4 \\(\\delta>0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(x\\in I\\)\uc774\uace0 \\(|x-c|<\\delta\\)\uc774\uba74
    \n\\[
    \n|f_{N+1}(x)-f_{N+1}(c)|<\\frac{\\varepsilon}{3}
    \n\\]
    \n\uc774\ub2e4. \ub530\ub77c\uc11c
    \n\\[
    \n\\begin{aligned}
    \n|f(x)-f(c)|
    \n&\\le |f(x)-f_{N+1}(x)|
    \n +|f_{N+1}(x)-f_{N+1}(c)|
    \n +|f_{N+1}(c)-f(c)|\\\\
    \n&<\\varepsilon.
    \n\\end{aligned}
    \n\\]
    \n\uadf8\ub7ec\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
    \n\\[
    \nf_n(x)=
    \n\\begin{cases}
    \nn & \\text{if }\\;0<x<\\dfrac1n,\\\\[5pt]
    \n0 & \\text{if }\\;x=0\\text{ or }x\\ge\\dfrac1n
    \n\\end{cases}
    \n\\]
    \n\uc774\ub77c\uace0 \uc815\uc758\ud558\uc790. \\(\\{f_n\\}\\)\uc758 \uadf9\ud55c\ud568\uc218\ub294 \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0\ub9cc
    \n\\[
    \n\\lim_{n\\to\\infty}\\int_0^1f_n(x)\\,dx
    \n\\]
    \n\uc640 \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 8.3. (\uade0\ub4f1\uc218\ub834\uacfc \uc801\ubd84\uc758 \uad00\uacc4)<\/span><\/p>\n

    \\(a<b\\)\uc774\uace0 \ubaa8\ub4e0 \\(f_n\\)\uc774 \\([a,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(f_n\\rightrightarrows f\\)\uc774\uba74 \\(f\\)\ub3c4 \\([a,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uace0
    \n\\[
    \n\\lim_{n\\to\\infty}\\int_a^b f_n(x)\\,dx
    \n=\\int_a^b f(x)\\,dx.\\tag{8.1}
    \n\\]<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/span>
    \n\uba3c\uc800 \\(f\\)\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ubcf4\uc774\uc790. \\(\\varepsilon>0\\)\uc744 \ud0dd\ud55c\ub2e4. \uade0\ub4f1\uc218\ub834\uc5d0 \uc758\ud574 \uc5b4\ub5a4 \\(N\\)\uc5d0 \ub300\ud558\uc5ec \ubaa8\ub4e0 \\(x\\in[a,b]\\)\uc5d0\uc11c
    \n\\[
    \n|f_{N+1}(x)-f(x)|<\\frac{\\varepsilon}{4(b-a)}
    \n\\]
    \n\uc774\ub2e4. \\(f_{N+1}\\)\uc774 \uc801\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \uc5b4\ub5a4 \ubd84\ud560 \\(P\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \nU(f_{N+1},P)-L(f_{N+1},P)<\\frac{\\varepsilon}{2}
    \n\\]
    \n\uc774\ub2e4. \uac01 \uc18c\uad6c\uac04\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\ud55c\uacfc \ud558\ud55c\uc740 \uac01\uac01 \\(f_{N+1}\\)\uc758 \uc0c1\ud55c\uacfc \ud558\ud55c\uc5d0\uc11c \\(\\varepsilon\/[4(b-a)]\\)\ub9cc\ud07c\ubc16\uc5d0 \ubc97\uc5b4\ub098\uc9c0 \uc54a\uc73c\ubbc0\ub85c
    \n\\[
    \nU(f,P)-L(f,P)
    \n<\\frac{\\varepsilon}{2}+\\frac{\\varepsilon}{2}
    \n=\\varepsilon.
    \n\\]
    \n\ub9ac\ub9cc \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \\(f\\)\ub294 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n

    \uc774\uc81c \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc744 \ud0dd\ud558\uc790. \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \ubaa8\ub4e0 \\(x\\in[a,b]\\)\uc5d0\uc11c
    \n\\[
    \n|f_n(x)-f(x)|<\\frac{\\varepsilon}{b-a}
    \n\\]
    \n\uc774\ubbc0\ub85c \uc801\ubd84\uc758 \uc808\ub313\uac12 \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud574
    \n\\[
    \n\\left|\\int_a^b f_n(x)\\,dx-\\int_a^b f(x)\\,dx\\right|
    \n\\le\\int_a^b|f_n(x)-f(x)|\\,dx
    \n<\\varepsilon.
    \n\\]
    \n\ub530\ub77c\uc11c \uc801\ubd84\uacfc \uadf9\ud55c\uc758 \uc21c\uc11c\ub97c \ubc14\uafc0 \uc218 \uc788\ub2e4.<\/span><\/p>\n<\/div>\n

    \n

    \uc815\ub9ac 8.4. (\uade0\ub4f1\uc218\ub834\uacfc \ubbf8\ubd84\uc758 \uad00\uacc4)<\/span><\/p>\n

    \\(a<b\\)\uc774\uace0 \ubaa8\ub4e0 \\(f_n\\)\uc774 \\([a,b]\\)\uc5d0\uc11c \\(C^1\\)\uc774\ub77c\uace0 \ud558\uc790. \uc5b4\ub5a4 \\(x_0\\in[a,b]\\)\uc5d0\uc11c \uc2e4\uc218\uc5f4 \\(\\{f_n(x_0)\\}\\)\uc774 \uc218\ub834\ud558\uace0, \ud568\uc218\uc5f4 \\(\\{f_n’\\}\\)\uc774 \\([a,b]\\)\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\{f_n\\}\\)\uc740 \\([a,b]\\)\uc5d0\uc11c \uc5b4\ub5a4 \ud568\uc218 \\(f\\)\uc5d0 \uade0\ub4f1\uc218\ub834\ud558\uace0, \\(f\\)\ub294 \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70
    \n\\[
    \nf'(x)=\\lim_{n\\to\\infty}f_n'(x)
    \n\\]
    \n\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/span>
    \n\\(f_n’\\rightrightarrows g\\)\ub77c\uace0 \ud558\uc790. \\(\\{f_n(x_0)\\}\\)\uc774 \ucf54\uc2dc \uc218\uc5f4\uc774\uace0 \\(\\{f_n’\\}\\)\uc774 \uade0\ub4f1 \ucf54\uc2dc\uc774\ubbc0\ub85c, \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \ucda9\ubd84\ud788 \ud070 \\(m,n\\)\uc774\uba74
    \n\\[
    \n|f_m(x_0)-f_n(x_0)|<\\frac{\\varepsilon}{2}.\\tag{8.2}
    \n\\]
    \n\ub610\ud55c \ubaa8\ub4e0 \\(t\\in[a,b]\\)\uc5d0\uc11c
    \n\\[
    \n|f_m'(t)-f_n'(t)|<\\frac{\\varepsilon}{2(b-a)}.\\tag{8.3}
    \n\\]
    \n\ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74 \uc784\uc758\uc758 \\(x\\in[a,b]\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \nf_m(x)-f_n(x)
    \n=f_m(x_0)-f_n(x_0)
    \n+\\int_{x_0}^{x}(f_m'(t)-f_n'(t))\\,dt.\\tag{8.4}
    \n\\]
    \n\ub530\ub77c\uc11c
    \n\\[
    \n|f_m(x)-f_n(x)|<\\varepsilon.\\tag{8.5}
    \n\\]
    \n\uade0\ub4f1 \ucf54\uc2dc \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \\(f_n\\rightrightarrows f\\)\uc778 \ud568\uc218 \\(f\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n

    \uac01 \\(f_n’\\)\uc774 \uc5f0\uc18d\uc774\uace0 \\(f_n’\\rightrightarrows g\\)\uc774\ubbc0\ub85c \uc815\ub9ac 8.2\uc5d0 \uc758\ud574 \\(g\\)\ub294 \uc5f0\uc18d\uc774\ub2e4. \ub610\ud55c
    \n\\[
    \nf_n(x)-f_n(x_0)=\\int_{x_0}^{x}f_n'(t)\\,dt
    \n\\]
    \n\uc5d0\uc11c \\(n\\to\\infty\\)\ub85c \ubcf4\ub0b4\uace0 \uc815\ub9ac 8.3\uc744 \uc801\uc6a9\ud558\uba74
    \n\\[
    \nf(x)-f(x_0)=\\int_{x_0}^{x}g(t)\\,dt.
    \n\\]
    \n\ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac\uc5d0 \uc758\ud574 \\(f’=g\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

    \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 8.5. (\ub514\ub2c8 \uc815\ub9ac)<\/span><\/p>\n

    \ud568\uc218\uc5f4 \\(\\{f_n\\}\\)\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 \\(\\{f_n\\}\\)\uc774 \uc5f0\uc18d\ud568\uc218 \\(f\\)\ub85c \uc810\ubcc4\uc218\ub834\ud558\uba74, \\(K\\)\uc5d0\uc11c \\(\\{f_n\\}\\)\uc740 \\(f\\)\ub85c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/span>
    \n\\(g_n(x)=f(x)-f_n(x)\\)\ub77c\uace0 \ud558\uc790. \uac01 \\(g_n\\)\uc740 \uc5f0\uc18d\ud568\uc218\uc758 \ucc28\uc774\ubbc0\ub85c \uc5f0\uc18d\uc774\ub2e4. \ub610\ud55c \\(f_n\\le f_{n+1}\\)\uc774\ubbc0\ub85c \\(g_{n+1}(x)\\le g_n(x)\\)\uc774\ub2e4. \uc810\ubcc4\uc218\ub834\uacfc \ub2e8\uc870\uc131\uc5d0 \uc758\ud574
    \n\\[
    \n0\\le g_{n+1}(x)\\le g_n(x)
    \n\\]
    \n\uc774\uace0 \uac01 \\(x\\in K\\)\uc5d0\uc11c \\(g_n(x)\\to0\\)\uc774\ub2e4.<\/p>\n

    \\(\\varepsilon>0\\)\uc774\ub77c\uace0 \ud558\uc790. \uac01 \\(x\\in K\\)\uc5d0 \ub300\ud574 \uc5b4\ub5a4 \uc790\uc5f0\uc218 \\(N_x\\)\uac00 \uc874\uc7ac\ud558\uc5ec
    \n\\[
    \ng_{N_x}(x)<\\frac{\\varepsilon}{2}
    \n\\]
    \n\uc774\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\\[
    \ng_{N_x}(y)<\\frac{3\\varepsilon}{4}
    \n\\]
    \n\uc774\ub2e4. \\(\\{U_x\\mid x\\in K\\}\\)\ub294 \\(K\\)\uc758 \uc5f4\ub9b0\ub36e\uac1c\uc774\uace0 \\(K\\)\uac00 \ucef4\ud329\ud2b8\uc774\ubbc0\ub85c
    \n\\[
    \nK\\subseteq U_{x_1}\\cup\\cdots\\cup U_{x_m}
    \n\\]
    \n\uc778 \uc720\ud55c\ubd80\ubd84\ub36e\uac1c\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(N=\\max\\{N_{x_1},\\ldots,N_{x_m}\\}\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

    \\(y\\in K\\)\uc774\uace0 \\(n>N\\)\uc774\uba74 \\(y\\in U_{x_i}\\)\uc778 \uc5b4\ub5a4 \\(i\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \ub530\ub77c\uc11c
    \n\\[
    \n0\\le g_n(y)\\le g_N(y)\\le g_{N_{x_i}}(y)
    \n<\\frac{3\\varepsilon}{4}
    \n<\\varepsilon.
    \n\\]
    \n\uc989 \ubaa8\ub4e0 \\(y\\in K\\)\uc5d0\uc11c \\(|f_n(y)-f(y)|<\\varepsilon\\)\uc774\ubbc0\ub85c \\(\\{f_n\\}\\)\uc740 \\(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\uc740 \ubd80\ubd84\ud569 \ud568\uc218\uc5f4\uc758 \uc218\ub834\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 8.6. (\ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4 \\(M\\)-\ud310\uc815\ubc95)<\/span><\/p>\n

    \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uc5f4 \\(\\{f_n\\}\\)\uacfc \uc74c\uc774 \uc544\ub2cc \uc2e4\uc218\uc5f4 \\(\\{M_n\\}\\)\uc774 \uc784\uc758\uc758 \\(n\\)\uacfc \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n|f_n(x)|\\le M_n
    \n\\]
    \n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\sum M_n\\)\uc774 \uc218\ub834\ud558\uba74 \\(\\sum f_n\\)\uc740 \\(I\\)\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/span>
    \n\\(m<n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n\\sup_{x\\in I}\\left|\\sum_{k=m}^{n}f_k(x)\\right|
    \n\\le\\sum_{k=m}^{n}M_k.
    \n\\]
    \n\uc218\ub834\uae09\uc218 \\(\\sum M_n\\)\uc758 \ucf54\uc2dc \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \uc6b0\ubcc0\uc740 \\(m,n\\to\\infty\\)\uc77c \ub54c \\(0\\)\uc73c\ub85c \uac08 \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \ubd80\ubd84\ud569 \ud568\uc218\uc5f4\uc740 \uade0\ub4f1 \ucf54\uc2dc \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\uace0, \uc815\ub9ac 8.1\uc5d0 \uc758\ud574 \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.<\/span><\/p>\n<\/div>\n

    \uc218\uc5f4 \\(\\{a_n\\}\\)\uacfc \uc2e4\uc218 \\(c\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n\\sum_{n=0}^{\\infty}a_n(x-c)^n\\tag{8.6}
    \n\\]
    \n\uaf34\uc758 \uae09\uc218\ub97c \uac70\ub4ed\uc81c\uacf1\uae09\uc218<\/span>(power series) \ub610\ub294 \uba71\uae09\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uace0, \\(c\\)\ub97c \uadf8 \uc911\uc2ec<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

    \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc218\ub834 \ubc94\uc704\ub294 \ub2e4\uc74c \uacf5\uc2dd\uc73c\ub85c \uc644\uc804\ud788 \uacb0\uc815\ub41c\ub2e4. \uc5ec\uae30\uc11c\ub294 \ube44\uc74c\uc218 \uc218\uc5f4\uc758 \uc0c1\uadf9\ud55c\uc5d0 \\(+\\infty\\)\ub3c4 \ud5c8\uc6a9\ud55c\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 8.7. (\ucf54\uc2dc-\uc544\ub2e4\ub9c8\ub974 \uacf5\uc2dd)<\/span><\/p>\n

    \uac70\ub4ed\uc81c\uacf1\uae09\uc218 \\(\\sum a_n(x-c)^n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n\\rho=\\varlimsup_{n\\to\\infty}|a_n|^{1\/n}\\in[0,\\infty]
    \n\\]
    \n\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uacfc \uac19\uc774 \\(R\\in[0,\\infty]\\)\ub97c \uc815\ud55c\ub2e4.
    \n\\[
    \nR=
    \n\\begin{cases}
    \n1\/\\rho & \\text{if }\\;0<\\rho<\\infty,\\\\[5pt]
    \n\\infty & \\text{if }\\;\\rho=0,\\\\[5pt]
    \n0 & \\text{if }\\;\\rho=\\infty.
    \n\\end{cases}
    \n\\]
    \n\uadf8\ub7ec\uba74 \\(|x-c|<R\\)\uc774\uba74 \uae09\uc218\ub294 \uc808\ub300\uc218\ub834\ud558\uace0, \\(|x-c|>R\\)\uc774\uba74 \ubc1c\uc0b0\ud55c\ub2e4. \uc774 \\(R\\)\uc744 \uc218\ub834\ubc18\uc9c0\ub984<\/span>(radius of convergence)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/span>
    \n\\(u_n=a_n(x-c)^n\\)\uc774\ub77c\uace0 \ud558\uba74
    \n\\[
    \n\\varlimsup_{n\\to\\infty}|u_n|^{1\/n}
    \n=|x-c|\\rho
    \n\\]
    \n\uc774\ub2e4. \\(|x-c|\\rho<1\\)\uc774\uba74 \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \\(\\sum u_n\\)\uc740 \uc808\ub300\uc218\ub834\ud55c\ub2e4. \ubc18\ub300\ub85c \\(|x-c|\\rho>1\\)\uc774\uba74 \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \uae09\uc218\ub294 \ubc1c\uc0b0\ud55c\ub2e4. \\(\\rho=0\\)\uacfc \\(\\rho=\\infty\\)\uc778 \uacbd\uc6b0\ub3c4 \uac19\uc740 \ud574\uc11d\uc73c\ub85c \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

    \ub530\ub77c\uc11c \uc218\ub834\ud558\ub294 \\(x\\)\ub4e4\uc758 \uc9d1\ud569\uc740 \uc911\uc2ec \\(c\\)\ub97c \ud3ec\ud568\ud558\ub294 \uad6c\uac04\uc774\ub2e4. \\(0<R<\\infty\\)\uc774\uba74 \\(|x-c|<R\\)\uc5d0\uc11c \ud56d\uc0c1 \uc808\ub300\uc218\ub834\ud558\uace0 \\(|x-c|>R\\)\uc5d0\uc11c \ubc1c\uc0b0\ud558\uba70, \ub450 \ub05d\uc810 \\(c-R\\), \\(c+R\\)\uc5d0\uc11c\uc758 \uc218\ub834 \uc5ec\ubd80\ub9cc \ubcc4\ub3c4\ub85c \ud655\uc778\ud558\uba74 \ub41c\ub2e4. \uc774\uc5d0 \ub530\ub77c \uc218\ub834\uad6c\uac04\uc740
    \n\\[
    \n[c-R,c+R],\\quad
    \n(c-R,c+R),\\quad
    \n[c-R,c+R),\\quad
    \n(c-R,c+R]
    \n\\]
    \n\uc911 \ud558\ub098\uc774\ub2e4. \\(R=0\\)\uc774\uba74 \uc911\uc2ec\uc5d0\uc11c\ub9cc \uc218\ub834\ud558\uace0, \\(R=\\infty\\)\uc774\uba74 \ubaa8\ub4e0 \uc2e4\uc218\uc5d0\uc11c \uc218\ub834\ud55c\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. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac{x^n}{n!}\\)<\/li>\n
    2. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac{x^n}{\\sqrt n}\\)<\/li>\n
    3. \\(\\displaystyle\\sum_{n=1}^{\\infty}n!\\,x^n\\)<\/li>\n
    4. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac{x^n}{2^n}\\)<\/li>\n
    5. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac{x^n}{n2^n}\\)<\/li>\n
    6. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac{3^nx^n}{n4^n}\\)<\/li>\n
    7. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac{x^n}{n^2+1}\\)<\/li>\n
    8. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac{x^n}{\\ln(n+1)}\\)<\/li>\n
    9. \\(\\displaystyle\\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 8.8. (\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 \uae09\uc218\ub294 \\([c,c+R]\\)\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(x=c-R\\)\uc5d0\uc11c \uc218\ub834\ud558\uba74 \\([c-R,c]\\)\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/span>
      \n\uc624\ub978\ucabd \ub05d\uc810\uc758 \uacbd\uc6b0\ub9cc \ubcf4\uc774\uba74 \ucda9\ubd84\ud558\ub2e4. \ubcc0\uc218 \\(x-c\\)\ub97c \ub2e4\uc2dc \\(x\\)\ub85c \uc4f0\uba74 \\(c=0\\)\uc774\ub77c \uac00\uc815\ud560 \uc218 \uc788\ub2e4. \\(b_n=a_nR^n\\)\uc774\ub77c\uace0 \ud558\uba74 \\(\\sum b_n\\)\uc774 \uc218\ub834\ud55c\ub2e4. \\(0\\le x\\le R\\)\uc5d0\uc11c \\(t=x\/R\\)\ub85c \ub193\uc73c\uba74 \\(0\\le t\\le1\\)\uc774\uace0 \\(\\sum b_nt^n\\)\uc758 \uade0\ub4f1\uc218\ub834\uc744 \ubcf4\uc774\uba74 \ub41c\ub2e4.<\/p>\n

      \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc744 \ud0dd\ud558\uc790. \\(\\sum b_n\\)\uc758 \ucf54\uc2dc \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \uc790\uc5f0\uc218 \\(N_0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(N>M>N_0\\)\uc774\uba74
      \n\\[
      \n\\left|\\sum_{n=M+1}^{N}b_n\\right|
      \n<\\frac{\\varepsilon}{2}
      \n\\]
      \n\uc774\ub2e4. \\(C_n=\\sum_{k=M+1}^{n}b_k\\)\ub77c\uace0 \ud558\uba74 \\(|C_n|<\\varepsilon\/2\\)\uc774\ub2e4. \uc544\ubca8\uc758 \ubd80\ubd84\ud569 \uacf5\uc2dd\uc5d0 \uc758\ud574 \\(0\\le t<1\\)\uc77c \ub54c
      \n\\[
      \n\\begin{aligned}
      \n\\left|\\sum_{n=M+1}^{N}b_nt^n\\right|
      \n&=\\left|C_Nt^N+\\sum_{n=M+1}^{N-1}C_n(t^n-t^{n+1})\\right|\\\\
      \n&\\le\\frac{\\varepsilon}{2}t^N
      \n+\\frac{\\varepsilon}{2}(1-t)\\sum_{n=M+1}^{N-1}t^n\\\\
      \n&<\\varepsilon.
      \n\\end{aligned}
      \n\\]
      \n\\(t=1\\)\uc5d0\uc11c\ub3c4 \ucc98\uc74c\uc758 \ucf54\uc2dc \ubd80\ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \ubd80\ubd84\ud569 \ud568\uc218\uc5f4\uc740 \\([0,1]\\)\uc5d0\uc11c \uade0\ub4f1 \ucf54\uc2dc\uc774\uace0 \uc815\ub9ac 8.1\uc5d0 \uc758\ud574 \uade0\ub4f1\uc218\ub834\ud55c\ub2e4. \uc67c\ucabd \ub05d\uc810\uc740 \\(x-c\\)\ub97c \\(-(x-c)\\)\ub85c \ubc14\uafb8\uba74 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

      \uade0\ub4f1\uc218\ub834\uc758 \uc131\uc9c8\uacfc \uc544\ubca8\uc758 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uba74 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc720\ud55c\ud569\uacfc \uac70\uc758 \uac19\uc740 \ubc29\uc2dd\uc73c\ub85c \ubbf8\ubd84\ud558\uace0 \uc801\ubd84\ud560 \uc218 \uc788\ub2e4.<\/p>\n

      \n

      \uc815\ub9ac 8.9. (\uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uae30\ubcf8\uc131\uc9c8)<\/span><\/p>\n

      \uac70\ub4ed\uc81c\uacf1\uae09\uc218
      \n\\[
      \nF(x)=\\sum_{n=0}^{\\infty}a_n(x-c)^n
      \n\\]
      \n\uc758 \uc218\ub834\ubc18\uc9c0\ub984\uc774 \\(R\\in(0,\\infty]\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

        \n
      1. \uc218\ub834\uad6c\uac04\uc5d0 \ud3ec\ud568\ub418\ub294 \ubaa8\ub4e0 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc5d0\uc11c \uae09\uc218\ub294 \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.<\/li>\n
      2. \\(|x-c|<R\\)\uc5d0\uc11c
        \n\\[
        \nF'(x)=\\sum_{n=1}^{\\infty}na_n(x-c)^{n-1}
        \n\\]
        \n\uc774\uace0, \uc624\ub978\ucabd\uc758 \ub3c4\ud568\uc218 \uae09\uc218\ub3c4 \uc218\ub834\ubc18\uc9c0\ub984 \\(R\\)\uc744 \uac00\uc9c4\ub2e4.<\/li>\n
      3. \\(a<b\\)\uac00 \uc218\ub834\uad6c\uac04\uc758 \uc810\uc774\uba74
        \n\\[
        \n\\int_a^bF(x)\\,dx
        \n=\\sum_{n=0}^{\\infty}a_n\\int_a^b(x-c)^n\\,dx.
        \n\\]<\/li>\n<\/ol>\n<\/div>\n
        \n

        \uc99d\uba85<\/span>
        \n\uba3c\uc800 \\(0<r<R\\)\ub77c\uace0 \ud558\uc790. \\(r<s<R\\)\uc778 \\(s\\)\ub97c \ud0dd\ud558\uba74 \\(\\sum|a_n|s^n\\)\uc740 \uc218\ub834\ud55c\ub2e4. \\(|x-c|\\le r\\)\uc5d0\uc11c
        \n\\[
        \n|a_n(x-c)^n|
        \n\\le |a_n|r^n
        \n\\le |a_n|s^n
        \n\\]
        \n\uc774\ubbc0\ub85c \ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4 \\(M\\)-\ud310\uc815\ubc95\uc5d0 \uc758\ud574 \\([c-r,c+r]\\)\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4. \uc218\ub834\uad6c\uac04\uc758 \ub05d\uc810\uc774 \ud3ec\ud568\ub418\ub294 \uacbd\uc6b0\uc5d0\ub294 \uc815\ub9ac 8.8\uc744 \ud568\uaed8 \uc801\uc6a9\ud558\uba74 \uadf8 \ub05d\uc810\uae4c\uc9c0 \uade0\ub4f1\uc218\ub834\ud55c\ub2e4. \ub530\ub77c\uc11c \uc218\ub834\uad6c\uac04 \uc548\uc758 \ubaa8\ub4e0 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.<\/p>\n

        \ub3c4\ud568\uc218 \uae09\uc218\uc758 \uc218\ub834\ubc18\uc9c0\ub984\uc744 \ubcf4\uc790. \\(n^{1\/n}\\to1\\)\uc774\ubbc0\ub85c
        \n\\[
        \n\\varlimsup_{n\\to\\infty}(n|a_n|)^{1\/n}
        \n=\\varlimsup_{n\\to\\infty}|a_n|^{1\/n}.
        \n\\]
        \n\ucf54\uc2dc-\uc544\ub2e4\ub9c8\ub974 \uacf5\uc2dd\uc5d0 \uc758\ud574 \\(\\sum_{n=1}^{\\infty}na_n(x-c)^n\\), \ub530\ub77c\uc11c \\(\\sum_{n=1}^{\\infty}na_n(x-c)^{n-1}\\)\ub3c4 \uc6d0\ub798 \uae09\uc218\uc640 \uac19\uc740 \uc218\ub834\ubc18\uc9c0\ub984 \\(R\\)\uc744 \uac00\uc9c4\ub2e4. \uc784\uc758\uc758 \\(r<R\\)\uc5d0 \ub300\ud558\uc5ec \ubd80\ubd84\ud569
        \n\\[
        \nF_N(x)=\\sum_{n=0}^Na_n(x-c)^n
        \n\\]
        \n\uacfc \uadf8 \ub3c4\ud568\uc218 \\(F_N’\\)\ub97c \\([c-r,c+r]\\)\uc5d0\uc11c \uc0dd\uac01\ud558\uc790. \uc55e\uc5d0\uc11c \ubcf4\uc778 \uade0\ub4f1\uc218\ub834\uc5d0 \uc758\ud574 \\(F_N’\\)\ub3c4 \uc774 \uad6c\uac04\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud558\uace0 \\(F_N(c)=a_0\\)\uc774\ub2e4. \uc815\ub9ac 8.4\uc5d0 \uc758\ud574 \\(F’=\\lim F_N’\\)\uc774\uace0 \ud56d\ubcc4\ubbf8\ubd84 \uacf5\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

        \ub9c8\uc9c0\ub9c9\uc73c\ub85c \\([a,b]\\)\uac00 \uc218\ub834\uad6c\uac04\uc5d0 \ud3ec\ud568\ub418\uba74 \uccab \ubc88\uc9f8 \uba85\uc81c\uc5d0 \uc758\ud574 \ubd80\ubd84\ud569\uc774 \\([a,b]\\)\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4. \uc815\ub9ac 8.3\uc744 \uc801\uc6a9\ud558\uba74 \ud56d\ubcc4\uc801\ubd84 \uacf5\uc2dd\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

        \n

        \ubb38\uc81c 8.6.<\/span>
        \n\ud568\uc218\uae09\uc218
        \n\\[
        \n\\sum_{n=1}^{\\infty}\\frac{x^n}{n}
        \n\\]
        \n\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n

          \n
        1. \uc784\uc758\uc758 \\(0<r<1\\)\uc5d0 \ub300\ud558\uc5ec \\([-r,r]\\)\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.<\/li>\n
        2. \\([0,1)\\)\uc5d0\uc11c\ub294 \uade0\ub4f1\uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
          \n

          \ubb38\uc81c 8.7.<\/span>
          \n\uc2e4\uc218\uc5f4 \\(\\{a_n\\}\\), \\(\\{b_n\\}\\)\uc5d0 \ub300\ud558\uc5ec
          \n\\[
          \n\\varlimsup_{n\\to\\infty}|a_n+b_n|^{1\/n}
          \n\\le
          \n\\max\\left\\{
          \n\\varlimsup_{n\\to\\infty}|a_n|^{1\/n},
          \n\\varlimsup_{n\\to\\infty}|b_n|^{1\/n}
          \n\\right\\}
          \n\\]
          \n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \uc774\ub97c \uc774\uc6a9\ud558\uc5ec \ub450 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc218\ub834\ubc18\uc9c0\ub984\uc774 \uac01\uac01 \\(R_1,R_2\\)\uc774\uba74 \ub450 \uae09\uc218\ub97c \ud56d\ubcc4\ub85c \ub354\ud55c \uae09\uc218\uc758 \uc218\ub834\ubc18\uc9c0\ub984\uc740 \uc801\uc5b4\ub3c4 \\(\\min\\{R_1,R_2\\}\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

          \n

          \ubb38\uc81c 8.8.<\/span>
          \n\ubb34\ud55c\ub4f1\ube44\uae09\uc218\uc640 \ud56d\ubcc4\ubbf8\ubd84\uc744 \uc774\uc6a9\ud558\uc5ec \\(|x|<1\\)\uc5d0\uc11c
          \n\\[
          \n\\frac1{(1-x)^2}
          \n=\\sum_{n=1}^{\\infty}nx^{n-1},
          \n\\quad
          \n\\frac{x}{(1-x)^2}
          \n=\\sum_{n=1}^{\\infty}nx^n
          \n\\]
          \n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

          \n

          \ubb38\uc81c 8.9.<\/span>
          \n\ud56d\ubcc4\uc801\ubd84\uc744 \uc774\uc6a9\ud558\uc5ec \\(|x|<1\\)\uc5d0\uc11c
          \n\\[
          \n\\ln\\frac{1+x}{1-x}
          \n=2\\sum_{n=0}^{\\infty}\\frac{x^{2n+1}}{2n+1}
          \n\\]
          \n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

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

          \uc815\uc758 5.8\uc758 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd<\/a>\uc5d0\uc11c \uc810 \\(c\\)\uc5d0\uc11c \\(n\\)\ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc758 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc744 \uc815\uc758\ud558\uc600\ub2e4. \uac19\uc740 \uc2dd\uc744 \ub2e4\uc2dc \uc4f0\uba74
          \n\\[
          \nP_n(x)=\\sum_{k=0}^{n}\\frac{f^{(k)}(c)}{k!}(x-c)^k.\\tag{8.7}
          \n\\]
          \n\ud568\uc218 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uc784\uc758 \ud69f\uc218\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c \ud615\uc2dd\uc801\uc778 \ubb34\ud55c\uae09\uc218
          \n\\[
          \n\\sum_{n=0}^{\\infty}\\frac{f^{(n)}(c)}{n!}(x-c)^n
          \n\\]
          \n\uc744 \\(c\\)\ub97c \uc911\uc2ec\uc73c\ub85c \ud55c \\(f\\)\uc758 \ud14c\uc77c\ub7ec \uae09\uc218<\/span>(Taylor series)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(c=0\\)\uc774\uba74 \ub9e5\ud074\ub77c\ub9b0 \uae09\uc218<\/span>(Maclaurin series)\ub77c\uace0 \ubd80\ub978\ub2e4. \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \uc2e4\uc81c\ub85c \\(f(x)\\)\uc5d0 \uc218\ub834\ud558\ub294\uc9c0\ub294 \ubcc4\ub3c4\uc758 \ubb38\uc81c\uc774\uba70,           \uc815\ub9ac 5.9\uc758 \ud14c\uc77c\ub7ec \uc815\ub9ac<\/a>\uc758 \ub098\uba38\uc9c0\uac00 \\(n\\to\\infty\\)\uc77c \ub54c \\(0\\)\uc73c\ub85c \uac00\ub294\uc9c0\ub97c \ud655\uc778\ud574\uc57c \ud55c\ub2e4.<\/p>\n

          \ub2e4\uc74c\uc740 \uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \uc804\uac1c\uc774\ub2e4. \uc2e4\uc218 \\(\\alpha\\)\uc640 \uc790\uc5f0\uc218 \\(n\\ge1\\)\uc5d0 \ub300\ud558\uc5ec
          \n\\[
          \n\\binom{\\alpha}{0}=1,\\quad
          \n\\binom{\\alpha}{n}
          \n=\\frac{\\alpha(\\alpha-1)\\cdots(\\alpha-n+1)}{n!}
          \n\\]
          \n\ub85c \ub454\ub2e4.<\/p>\n

          \n

          \uc815\ub9ac 8.10. (\uae30\ubcf8\uc801\uc778 \uac70\ub4ed\uc81c\uacf1\uae09\uc218 \uc804\uac1c)<\/span><\/p>\n

          \ub2e4\uc74c \ub4f1\uc2dd\uc774 \uac01 \ud45c\uc2dc\ub41c \ubc94\uc704\uc5d0\uc11c \uc131\ub9bd\ud55c\ub2e4.
          \n\\[
          \n\\begin{aligned}
          \ne^x&=\\sum_{n=0}^{\\infty}\\frac{x^n}{n!}
          \n&& (x\\in\\mathbb R),\\\\[4pt]
          \n\\sin x&=\\sum_{n=0}^{\\infty}\\frac{(-1)^nx^{2n+1}}{(2n+1)!}
          \n&& (x\\in\\mathbb R),\\\\[4pt]
          \n\\cos x&=\\sum_{n=0}^{\\infty}\\frac{(-1)^nx^{2n}}{(2n)!}
          \n&& (x\\in\\mathbb R),\\\\[4pt]
          \n\\frac1{1-x}&=\\sum_{n=0}^{\\infty}x^n
          \n&& (|x|<1),\\\\[4pt]
          \n\\ln(1+x)&=\\sum_{n=1}^{\\infty}\\frac{(-1)^{n-1}x^n}{n}
          \n&& (-1<x\\le1),\\\\[4pt]
          \n(1+x)^\\alpha&=\\sum_{n=0}^{\\infty}\\binom{\\alpha}{n}x^n
          \n&& (|x|<1).
          \n\\end{aligned}
          \n\\]<\/p>\n<\/div>\n

          \n

          \uc99d\uba85<\/span>
          \n\\(e^x\\), \\(\\sin x\\), \\(\\cos x\\)\uc5d0\ub294 \uc815\ub9ac 5.9\ub97c \uc801\uc6a9\ud55c\ub2e4. \\(x<0\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(t\\mapsto f(-t)\\)\uc5d0 \uac19\uc740 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74 \uc911\uc2ec\uc774 \\(0\\)\uc778 \ub3d9\uc77c\ud55c \ub098\uba38\uc9c0 \uacf5\uc2dd\uc744 \uc5bb\ub294\ub2e4. \uace0\uc815\ub41c \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uc9c0\uc218\ud568\uc218\uc758 \ub098\uba38\uc9c0\ub294 \uc5b4\ub5a4 \\(\\xi\\)\uac00 \\(0\\)\uacfc \\(x\\) \uc0ac\uc774\uc5d0 \uc874\uc7ac\ud558\uc5ec
          \n\\[
          \n|R_n(x)|
          \n\\le e^{|x|}\\frac{|x|^{n+1}}{(n+1)!}
          \n\\]
          \n\uc774\uace0, \ube44 \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \uc6b0\ubcc0\uc740 \\(0\\)\uc73c\ub85c \uc218\ub834\ud55c\ub2e4. \uc0ac\uc778\uacfc \ucf54\uc0ac\uc778\uc758 \ubaa8\ub4e0 \ub3c4\ud568\uc218\uc758 \uc808\ub313\uac12\uc740 \\(1\\) \uc774\ud558\uc774\ubbc0\ub85c \ub098\uba38\uc9c0\ub294 \\(|x|^{n+1}\/(n+1)!\\) \uc774\ud558\uc774\ub2e4. \ub530\ub77c\uc11c \ucc98\uc74c \uc138 \uc804\uac1c\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

          \ub124 \ubc88\uc9f8 \uc2dd\uc740 \uc815\ub9ac 7.1\uc758 \uae30\ud558\uae09\uc218 \uacf5\uc2dd<\/a>\uc774\ub2e4. \\(|x|<1\\)\uc5d0\uc11c
          \n\\[
          \n\\frac1{1+t}=\\sum_{n=0}^{\\infty}(-1)^nt^n
          \n\\]
          \n\uc744 \\(0\\)\uc5d0\uc11c \\(x\\)\uae4c\uc9c0 \ud56d\ubcc4\uc801\ubd84\ud558\uba74
          \n\\[
          \n\\ln(1+x)=\\sum_{n=1}^{\\infty}\\frac{(-1)^{n-1}x^n}{n}
          \n\\]
          \n\uc744 \uc5bb\ub294\ub2e4. \\(x=1\\)\uc5d0\uc11c\ub294 \uad50\ub300\uae09\uc218 \ud310\uc815\ubc95\uacfc \uc815\ub9ac 8.8\uc744 \uc801\uc6a9\ud558\uba74 \uac19\uc740 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

          \ub9c8\uc9c0\ub9c9\uc73c\ub85c
          \n\\[
          \nS(x)=\\sum_{n=0}^{\\infty}\\binom{\\alpha}{n}x^n
          \n\\]
          \n\uc774\ub77c\uace0 \ud558\uc790. \\(\\alpha\\)\uac00 \uc74c\uc774 \uc544\ub2cc \uc815\uc218\uc774\uba74 \uc774 \uae09\uc218\ub294 \uc720\ud55c\ud569\uc774\uace0, \uadf8\ub807\uc9c0 \uc54a\uc73c\uba74 \uc5f0\uc18d\ud56d\uc758 \uacc4\uc218\ube44\uac00 \uc808\ub313\uac12\uc73c\ub85c \\(1\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \uc218\ub834\ubc18\uc9c0\ub984\uc740 \\(1\\)\uc774\ub2e4. \\(|x|<1\\)\uc5d0\uc11c \ud56d\ubcc4\ubbf8\ubd84\uc744 \ud560 \uc218 \uc788\uace0
          \n\\[
          \n(n+1)\\binom{\\alpha}{n+1}
          \n=(\\alpha-n)\\binom{\\alpha}{n}
          \n\\]
          \n\uc774\ubbc0\ub85c
          \n\\[
          \n(1+x)S'(x)=\\alpha S(x).
          \n\\]
          \n\ub530\ub77c\uc11c
          \n\\[
          \n\\frac{d}{dx}\\bigl((1+x)^{-\\alpha}S(x)\\bigr)=0.
          \n\\]
          \n\\(S(0)=1\\)\uc774\ubbc0\ub85c \\(S(x)=(1+x)^\\alpha\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

          \ud568\uc218 \\(f\\)\uac00 \uc810 \\(c\\)\uc758 \uc5b4\ub5a4 \uc5f4\ub9b0\uadfc\ubc29\uc5d0\uc11c \\(c\\)\ub97c \uc911\uc2ec\uc73c\ub85c \ud55c \uc790\uc2e0\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uc640 \uc77c\uce58\ud560 \ub54c \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uc2e4\ud574\uc11d\uc801<\/span>\uc774\ub2e4(real analytic)\ub77c\uace0 \ud55c\ub2e4. \uc5f4\ub9b0\uad6c\uac04 \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \uc2e4\ud574\uc11d\uc801\uc774\uba74 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc2e4\ud574\uc11d\uc801\uc774\ub77c\uace0 \ud55c\ub2e4.<\/p>\n

          \uc784\uc758 \ud69f\uc218\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\ub294 \uc870\uac74\ub9cc\uc73c\ub85c\ub294 \uc2e4\ud574\uc11d\uc131\uc744 \uc5bb\uc744 \uc218 \uc5c6\ub2e4. \ub2e4\uc74c \uc608\uac00 \uc774\ub97c \ubcf4\uc5ec \uc900\ub2e4.<\/p>\n

          \n

          \uc815\ub9ac 8.11. (\ub9e4\ub044\ub7fd\uc9c0\ub9cc \ud574\uc11d\uc801\uc774\uc9c0 \uc54a\uc740 \ud568\uc218)<\/span><\/p>\n

          \ud568\uc218
          \n\\[
          \nf(x)=
          \n\\begin{cases}
          \ne^{-1\/x^2} & \\text{if }\\;x\\ne0,\\\\[5pt]
          \n0 & \\text{if }\\;x=0
          \n\\end{cases}
          \n\\]
          \n\uc740 \\(\\mathbb R\\)\uc5d0\uc11c \\(C^\\infty\\)\uc774\uace0 \ubaa8\ub4e0 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(f^{(n)}(0)=0\\)\uc774\uc9c0\ub9cc, \\(0\\)\uc5d0\uc11c \uc2e4\ud574\uc11d\uc801\uc774\uc9c0 \uc54a\ub2e4.<\/p>\n<\/div>\n

          \n

          \uc99d\uba85<\/span>
          \n\uba3c\uc800 \ubaa8\ub4e0 \uc2e4\uc218 \\(m>0\\)\uc5d0 \ub300\ud558\uc5ec
          \n\\[
          \n\\lim_{x\\to0}|x|^{-m}e^{-1\/x^2}=0
          \n\\]
          \n\uc784\uc744 \ubcf4\uc790. \\(t=1\/x^2\\)\ub85c \ub193\uace0 \\(N>m\/2\\)\uc778 \uc790\uc5f0\uc218 \\(N\\)\uc744 \ud0dd\ud558\uba74 \\(e^t\\ge t^N\/N!\\)\uc774\ubbc0\ub85c
          \n\\[
          \nt^{m\/2}e^{-t}
          \n\\le N!\\,t^{m\/2-N}
          \n\\to0.
          \n\\]
          \n\\(x\\ne0\\)\uc5d0\uc11c \\(f^{(n)}(x)\\)\ub294 \uc5b4\ub5a4 \ub2e4\ud56d\uc2dd \\(P_n\\)\uc5d0 \ub300\ud558\uc5ec
          \n\\[
          \nf^{(n)}(x)=P_n(1\/x)e^{-1\/x^2}
          \n\\]
          \n\uaf34\uc774\ub2e4. \uc774\ub97c \uadc0\ub0a9\uc801\uc73c\ub85c \ubbf8\ubd84\ud558\uba74 \uac19\uc740 \ud615\ud0dc\uac00 \uc720\uc9c0\ub41c\ub2e4. \uc704 \uadf9\ud55c\uc744 \uc0ac\uc6a9\ud558\uba74 \uac01 \ub2e8\uacc4\uc5d0\uc11c
          \n\\[
          \n\\lim_{h\\to0}\\frac{f^{(n)}(h)-f^{(n)}(0)}{h}=0
          \n\\]
          \n\uc774\ubbc0\ub85c \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(f^{(n)}(0)=0\\)\uc774\ub2e4. \uac19\uc740 \uadf9\ud55c\uc740 \\(f^{(n)}(x)\\to0=f^{(n)}(0)\\)\ub3c4 \ubcf4\uc5ec \uc8fc\ubbc0\ub85c \ubaa8\ub4e0 \ub3c4\ud568\uc218\ub294 \\(0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \ub530\ub77c\uc11c \\(f\\in C^\\infty\\)\uc774\uace0 \\(0\\)\uc5d0\uc11c\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\ub294 \ud56d\ub4f1\uc801\uc73c\ub85c \\(0\\)\uc774\uc9c0\ub9cc \\(x\\ne0\\)\uc774\uba74 \\(f(x)>0\\)\uc774\ubbc0\ub85c \\(f\\)\ub294 \\(0\\)\uc5d0\uc11c \uc2e4\ud574\uc11d\uc801\uc774\uc9c0 \uc54a\ub2e4.<\/span><\/p>\n<\/div>\n

          \uc2e4\ud574\uc11d\uc801 \ud568\uc218\uc758 \uc911\uc694\ud55c \ud2b9\uc9d5\uc740 \ud55c \uc810 \uadfc\ucc98\uc758 \uac70\ub4ed\uc81c\uacf1\uae09\uc218 \uc815\ubcf4\uac00 \ud568\uc218 \uc790\uccb4\ub97c \uac15\ud558\uac8c \uacb0\uc815\ud55c\ub2e4\ub294 \uc810\uc774\ub2e4.<\/p>\n

          \n

          \uc815\ub9ac 8.12. (\uc2e4\ud574\uc11d\ud568\uc218\uc758 \ud56d\ub4f1\uc815\ub9ac)<\/span><\/p>\n

          \uc5f4\ub9b0\uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc2e4\ud574\uc11d\uc801\uc778 \ud568\uc218 \\(f\\)\uc758 \uc601\uc810\ub4e4\uc774 \\(I\\) \uc548\uc5d0 \uc9d1\uc801\uc810\uc744 \uac00\uc9c0\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \ud56d\ub4f1\uc801\uc73c\ub85c \\(0\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(f\\)\uac00 \ud56d\ub4f1\uc801\uc73c\ub85c \\(0\\)\uc774 \uc544\ub2c8\uba74 \ubaa8\ub4e0 \uc601\uc810\uc740 \uace0\ub9bd\ub418\uc5b4 \uc788\ub2e4.<\/p>\n<\/div>\n

          \n

          \uc99d\uba85<\/span>
          \n\uc11c\ub85c \ub2e4\ub978 \uc601\uc810 \\(x_n\\)\uc774 \\(c\\in I\\)\ub85c \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uc5f0\uc18d\uc131\uc5d0 \uc758\ud574 \\(f(c)=0\\)\uc774\ub2e4. \ub864\uc758 \uc815\ub9ac\ub97c \ubc18\ubcf5\ud558\uc5ec \uc801\uc6a9\ud558\uba74 \uac01 \uc790\uc5f0\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(c\\)\ub85c \uc218\ub834\ud558\ub294 \\(f^{(k)}\\)\uc758 \uc601\uc810 \uc218\uc5f4\uc744 \uc5bb\uc744 \uc218 \uc788\uace0, \ub530\ub77c\uc11c \uc5f0\uc18d\uc131\uc5d0 \uc758\ud574 \\(f^{(k)}(c)=0\\)\uc774\ub2e4. \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uc2e4\ud574\uc11d\uc801\uc774\ubbc0\ub85c \\(c\\)\uc758 \uc5b4\ub5a4 \uadfc\ubc29\uc5d0\uc11c \\(f=0\\)\uc774\ub2e4.<\/p>\n

          \uc774\uc81c
          \n\\[
          \nA=\\{x\\in I\\mid x\\text{\uc758 \uc5b4\ub5a4 \uadfc\ubc29\uc5d0\uc11c }f=0\\}
          \n\\]
          \n\ub77c\uace0 \ud558\uc790. \uc55e\uc758 \ub17c\uc758\uc5d0\uc11c \\(A\\ne\\varnothing\\)\uc774\uace0 \uc815\uc758\uc0c1 \\(A\\)\ub294 \uc5f4\ub824 \uc788\ub2e4. \\(x_j\\in A\\)\uc774\uace0 \\(x_j\\to x\\in I\\)\uc774\uba74 \ubaa8\ub4e0 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(f^{(k)}(x_j)=0\\)\uc774\uace0, \ub3c4\ud568\uc218\uc758 \uc5f0\uc18d\uc131\uc5d0 \uc758\ud574 \\(f^{(k)}(x)=0\\)\uc774\ub2e4. \\(f\\)\uac00 \\(x\\)\uc5d0\uc11c \uc2e4\ud574\uc11d\uc801\uc774\ubbc0\ub85c \\(x\\in A\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(A\\)\ub294 \\(I\\)\uc5d0\uc11c \ub2eb\ud600 \uc788\ub2e4. \uad6c\uac04 \\(I\\)\uac00 \uc5f0\uacb0\ub418\uc5b4 \uc788\uc73c\ubbc0\ub85c \\(A=I\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

          \uc2e4\ud574\uc11d\uc801 \ud568\uc218\uc758 \ud569\uc774 \uc2e4\ud574\uc11d\uc801\uc774\ub77c\ub294 \uc0ac\uc2e4\uc740 \uc815\uc758\uc5d0\uc11c \ubc14\ub85c \ub530\ub978\ub2e4. \uacf1\uc758 \uacbd\uc6b0\ub294 \ubb38\uc81c 8.14\uc5d0\uc11c \ucf54\uc2dc \uacf1\uc744 \uc774\uc6a9\ud558\uc5ec \ud655\uc778\ud55c\ub2e4. \ud569\uc131\ub3c4 \uc2e4\ud574\uc11d\uc131\uc744 \ubcf4\uc874\ud558\uc9c0\ub9cc, \ud569\uc131\uae09\uc218\uc758 \uc808\ub300\uc218\ub834\uacfc \uc7ac\ubc30\uc5f4\uc744 \ub2e4\ub8e8\ub294 \uc99d\uba85\uc740 \uc5ec\uae30\uc11c\ub294 \uc0dd\ub7b5\ud55c\ub2e4. \ub610\ud55c \ubcf5\uc18c\ud574\uc11d\ud559\uc5d0\uc11c\ub294 \uc2e4\ud574\uc11d\uc801 \ud568\uc218\uc758 \uad6d\uc18c\uc801 \ubcf5\uc18c \ud655\uc7a5\uacfc \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc218\ub834\ubc18\uc9c0\ub984\uc744 \ubcf5\uc18c \ud2b9\uc774\uc810\uacfc \uc5f0\uacb0\ud558\uc5ec \uc124\uba85\ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4
          \n\\[
          \n\\frac1{1+x^2}
          \n=\\sum_{n=0}^{\\infty}(-1)^nx^{2n}
          \n\\quad(|x|<1)
          \n\\]
          \n\uc758 \uc218\ub834\ubc18\uc9c0\ub984\uc774 \\(1\\)\uc778 \ud604\uc0c1\uc740 \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \\(\\pm i\\)\uac00 \ubd84\ubaa8\uc758 \uc601\uc810\uc774\ub77c\ub294 \uc0ac\uc2e4\uacfc \uad00\ub828\ub41c\ub2e4. \uc774 \uad00\uc810\uc740 \ubcf5\uc18c\ud574\uc11d\ud559\uc5d0\uc11c \ub2e4\ub8ec\ub2e4.<\/p>\n

          \n

          \ubb38\uc81c 8.10.<\/span>
          \n\ud568\uc218 \\(f\\)\uac00 \uc810 \\(c\\)\uc5d0\uc11c \uc2e4\ud574\uc11d\uc801\uc77c \ub54c, \\(c\\)\ub97c \uc911\uc2ec\uc73c\ub85c \ud558\uace0 \\(c\\)\uc758 \uc5b4\ub5a4 \uadfc\ubc29\uc5d0\uc11c \\(f\\)\uc5d0 \uc218\ub834\ud558\ub294 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uac00 \uc720\uc77c\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

          \n

          \ubb38\uc81c 8.11.<\/span>
          \n\ud568\uc218
          \n\\[
          \nf(x)=\\frac1{(1-x)^2}
          \n\\]
          \n\uc758 \ub9e5\ud074\ub77c\ub9b0 \uae09\uc218\ub97c \uad6c\ud558\uace0 \uc218\ub834\ubc18\uc9c0\ub984\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

          \n

          \ubb38\uc81c 8.12.<\/span>
          \n\uad6c\uac04 \\([-1,1]\\)\uc5d0\uc11c \\(f(x)=\\tan^{-1}x\\)\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n

            \n
          1. \ubb34\ud55c\ub4f1\ube44\uae09\uc218\uc758 \ud569\uc744 \uc0ac\uc6a9\ud558\uc5ec \\(-1<x<1\\)\uc5d0\uc11c \\(\\dfrac1{1+x^2}\\)\uc744 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/li>\n
          2. \\(f'(x)\\)\ub97c \uad6c\ud558\uace0 (1)\uc758 \uacb0\uacfc\ub85c\ubd80\ud130 \\(-1<x<1\\)\uc5d0\uc11c \\(f'(x)\\)\ub97c \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/li>\n
          3. \ud56d\ubcc4\uc801\ubd84\uc744 \uc0ac\uc6a9\ud558\uc5ec \\(-1<x<1\\)\uc5d0\uc11c
            \n\\[
            \n\\tan^{-1}x
            \n=\\sum_{n=0}^{\\infty}\\frac{(-1)^nx^{2n+1}}{2n+1}
            \n\\]
            \n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n
          4. \uc544\ubca8\uc758 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec
            \n\\[
            \n\\frac{\\pi}{4}
            \n=1-\\frac13+\\frac15-\\frac17+\\cdots
            \n\\]
            \n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \uc774 \uacf5\uc2dd\uc744 \uc6d0\uc8fc\uc728\uc5d0 \ub300\ud55c \ub77c\uc774\ud504\ub2c8\uce20 \uacf5\uc2dd<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
            \n

            \ubb38\uc81c 8.13.<\/span>
            \n\ubb34\ud55c\uae09\uc218
            \n\\[
            \n\\sum_{n=1}^{\\infty}\\frac{\\sin n}{n}
            \n\\]
            \n\uc774 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc2dc\uc624. (\uc0bc\uac01\ud568\uc218\uc758 \ub367\uc148\uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uc5ec \\(\\sum_{k=1}^N\\sin k\\)\uac00 \\(N\\)\uacfc \ubb34\uad00\ud558\uac8c \uc720\uacc4\uc784\uc744 \ubcf4\uc778 \ub4a4 \ub514\ub9ac\ud074\ub808 \ud310\uc815\ubc95\uc744 \uc801\uc6a9\ud55c\ub2e4.)<\/p>\n<\/div>\n

            \n

            \ubb38\uc81c 8.14.<\/span>
            \n\uac70\ub4ed\uc81c\uacf1\uae09\uc218 \\(\\sum_{n=0}^{\\infty}a_nx^n\\)\uacfc \\(\\sum_{n=0}^{\\infty}b_nx^n\\)\uc774 \\((-R,R)\\)\uc5d0\uc11c \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \\(n\\ge0\\)\uc5d0 \ub300\ud558\uc5ec
            \n\\[
            \nc_n=\\sum_{j=0}^{n}a_jb_{n-j}
            \n\\]
            \n\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(x\\in(-R,R)\\)\uc5d0 \ub300\ud558\uc5ec \\(\\sum c_nx^n\\)\uc774 \uc808\ub300\uc218\ub834\ud558\uace0
            \n\\[
            \n\\left(\\sum_{n=0}^{\\infty}a_nx^n\\right)
            \n\\left(\\sum_{n=0}^{\\infty}b_nx^n\\right)
            \n=\\sum_{n=0}^{\\infty}c_nx^n
            \n\\]
            \n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \uc6b0\ubcc0\uc744 \ub450 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \ucf54\uc2dc \uacf1<\/span>(Cauchy product)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n

            \n

            \ud574\uc11d\ud559 \uac15\uc758\ub178\ud2b8<\/a><\/p>\n

              \n
            1. \uc2e4\uc218\uacc4\uc758 \uc131\uc9c8<\/a><\/li>\n
            2. \uac70\ub9ac\uacf5\uac04<\/a><\/li>\n
            3. \uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \uc704\uc0c1\uc801 \uc131\uc9c8<\/a><\/li>\n
            4. \ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131<\/a><\/li>\n
            5. \uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n
            6. \uc77c\ubcc0\uc218 \ud568\uc218\uc758 \uc801\ubd84<\/a><\/li>\n
            7. \ubb34\ud55c\uae09\uc218<\/a><\/li>\n
            8. \uc2e4\ud574\uc11d\uc801 \ud568\uc218<\/a><\/li>\n
            9. \ub2e4\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n
            10. \uc911\uc801\ubd84<\/a><\/li>\n
            11. \ubca1\ud130\uc7a5\uacfc \uc801\ubd84 \uc815\ub9ac<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n

              <\/p>\n","protected":false},"excerpt":{"rendered":"

              \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. \ud568\uc218\uc5f4\uc758 \uade0\ub4f1\uc218\ub834 \uac01 \ud56d\uc774 \ud568\uc218\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc218\uc5f4\uc744 \ud568\uc218\uc5f4(sequence of functions)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\{f_n\\}\\)\uc774 \uc9d1\ud569 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc2e4\ud568\uc218\uc758 \ud568\uc218\uc5f4\uc774\uace0 \\(f\\colon I\\to\\mathbb R\\)\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\( \\lim_{n\\to\\infty}f_n(x)=f(x) \\) \uc774\uba74 \\(I\\)\uc5d0\uc11c \\(\\{f_n\\}\\)\uc774 \\(f\\)\uc5d0 \uc810\ubcc4\uc218\ub834\ud55c\ub2e4(converges pointwise)\uace0 \ud55c\ub2e4. \uc774\ub54c \\(f\\)\ub97c \uc810\ubcc4\uadf9\ud55c\ud568\uc218(pointwise limit function) \ub610\ub294 \uac04\ub2e8\ud788 \uadf9\ud55c\ud568\uc218(limit function)\ub77c\uace0 \ubd80\ub974\uace0 \\(f_n\\to f\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ubb38\uc81c 8.1. \\(f_n(x)=x^n\\)\uc77c \ub54c \\([0,1]\\)\uc5d0\uc11c…<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9470,"menu_order":108,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9493","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9493","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=9493"}],"version-history":[{"count":11,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9493\/revisions"}],"predecessor-version":[{"id":10113,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9493\/revisions\/10113"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9470"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9493"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}