{"id":2024,"date":"2019-06-26T13:17:04","date_gmt":"2019-06-26T04:17:04","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=2024"},"modified":"2020-05-14T10:31:44","modified_gmt":"2020-05-14T01:31:44","slug":"calculus-taylor-series-and-maclaurin-series","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-taylor-series-and-maclaurin-series\/","title":{"rendered":"\ud14c\uc77c\ub7ec \uae09\uc218"},"content":{"rendered":"<p>\uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uacf5\ubd80\ud560 \ub54c \uc911\uc810\uc801\uc73c\ub85c \uc0b4\ud3b4\ubcf4\uc544\uc57c \ud560 \ub0b4\uc6a9\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc138 \uac00\uc9c0\uc774\ub2e4.<\/p>\n<ul>\n<li>\uac70\ub4ed\uc81c\uacf1\uae09\uc218 \\(\\sum a_n (x-c)^n \\)\uc774 \uc218\ub834\ud558\ub3c4\ub85d \ud558\ub294 \\(x\\)\uc758 \uac12(\ubc94\uc704)\uc744 \uc5b4\ub5bb\uac8c \uad6c\ud560 \uac83\uc778\uac00?<\/li>\n<li>\uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \uc815\uc758\ub41c \ud568\uc218\ub97c \ubbf8\ubd84\ud558\uac70\ub098 \uc801\ubd84\ud560 \ub550 \uc5b4\ub5bb\uac8c \ud558\ub294\uac00?<\/li>\n<li>\uc5b4\ub5a0\ud55c \ud568\uc218\ub97c \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub294\uac00?<\/li>\n<\/ul>\n<p>\uc774 \uc911 \uc138 \ubc88\uc9f8 \uc9c8\ubb38\uc5d0 \ub300\ud55c \ub2f5\uc774 \ubc14\ub85c \ud14c\uc77c\ub7ec \uae09\uc218\uc640 \ud14c\uc77c\ub7ec\uc758 \uc815\ub9ac\uc774\ub2e4. \ud14c\uc77c\ub7ec \uae09\uc218\ub294 \uc8fc\uc5b4\uc9c4 \ud568\uc218 \\(f\\)\ub97c \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \ub098\ud0c0\ub0b4\ub294 \ubc29\ubc95\uc744 \uc81c\uacf5\ud55c\ub2e4. \uc989 \ud568\uc218 \\(f\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c \uc774 \ud568\uc218\ub85c\ubd80\ud130 \uac70\ub4ed\uc81c\uacf1\uae09\uc218 \\(\\sum a_n (x-c)^n\\)\uc744 \uad6c\ud558\ub294 \ubc29\ubc95\uc744 \uc81c\uacf5\ud55c\ub2e4. \uadf8\ub7ec\ub098 \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uad6c\ud588\ub2e4\uace0 \ud574\uc11c \uadf8 \uae09\uc218\uac00 \uc6d0\ub798\uc758 \ud568\uc218\uc640 \uc77c\uce58\ud55c\ub2e4\ub294 \uac83\uc740 \ubcf4\uc7a5\ud560 \uc218 \uc5c6\ub2e4. \uc774\ub54c \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \uc6d0\ub798\uc758 \ud568\uc218\ub85c \uc218\ub834\ud558\ub294\uc9c0 \uc5ec\ubd80\ub97c \ubc1d\ud788\ub294 \ubc29\ubc95\uc774 \ud14c\uc77c\ub7ec\uc758 \uc815\ub9ac\uc774\ub2e4.<\/p>\n<p>\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ud14c\uc77c\ub7ec\uc758 \uae09\uc218\uc640 \ud14c\uc77c\ub7ec\uc758 \uc815\ub9ac\ub97c \uc0b4\ud3b4\ubcf4\uace0, \ud14c\uc77c\ub7ec \uae09\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub294 \ud568\uc218\uc758 \uc608\ub97c \ud568\uaed8 \uc0b4\ud3b4\ubcf4\uc790. \uc989 \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \ub450 \uac00\uc9c0 \uc9c8\ubb38\uc5d0 \ub300\ud55c \ub2f5\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<ul>\n<li>\ud568\uc218 \\(f\\)\uac00 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \ud45c\ud604\ub420 \uc218 \uc788\ub2e4\uba74, \uadf8 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc5b4\ub5bb\uac8c \uad6c\ud560 \uac83\uc778\uac00?<\/li>\n<li>\uad6c\ud55c \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uac00 \\(f\\)\uc640 \uc77c\uce58\ud55c\ub2e4\ub294 \uac83\uc744 \uc5b4\ub5bb\uac8c \ubcf4\uc77c \uac83\uc778\uac00?<\/li>\n<\/ul>\n<p>\ud45c\uae30\uc758 \ud3b8\uc758\uc0c1 \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c \ud14c\uc77c\ub7ec \uae09\uc218\uc758 \uc911\uc2ec\uc740 \\(c\\)\uac00 \uc544\ub2cc \\(a\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub85c \ud55c\ub2e4.<\/p>\n<div class=\"box contentindex\">\n<h3 class=\"indextitle\">\ub0b4\uc6a9 \uc21c\uc11c<\/h3>\n<ul>\n<li><a href=\"#taylorseries\">\ud14c\uc77c\ub7ec \uae09\uc218<\/a><\/li>\n<li><a href=\"#uniqueness\">\ud14c\uc77c\ub7ec \uae09\uc218\uc758 \uc720\uc77c\uc131<\/a><\/li>\n<li><a href=\"#convergence\">\ud14c\uc77c\ub7ec \uae09\uc218\uc758 \uc218\ub834\uc131<\/a><\/li>\n<li><a href=\"#approximation\">\uc624\ucc28 \ucd94\uc815<\/a><\/li>\n<li><a href=\"#nonanalyticfunctions\">\ud14c\uc77c\ub7ec \uae09\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc5c6\ub294 \ud568\uc218\uc758 \uc608<\/a><\/li>\n<\/ul>\n<h3 class=\"pretitle\">\ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9<\/h3>\n<ul>\n<li>\uac70\ub4ed\uc81c\uacf1\uae09\uc218 (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Power_series\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \ud310\uc815 (<a href=\"\/blog\/articles\/calculus-convergence-tests-of-series\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\ud3c9\uade0\uac12 \uc815\ub9ac (<a href=\"\/blog\/articles\/calculus-the-mean-value-theorem\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\uc0bc\uac01\ud568\uc218 (<a href=\"\/blog\/articles\/calculus-derivatives-of-trigonometric-functions\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218 (<a href=\"\/blog\/articles\/calculus-derivatives-of-exponential-and-logarithm-functions\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<\/ul>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"taylorseries\"><\/a><\/p>\n<h3>\ud14c\uc77c\ub7ec \uae09\uc218<\/h3>\n<p>\ud568\uc218 \\(f\\)\uac00 \\(a\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04 \\((a-R ,\\, a+R )\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0, \uc774 \uad6c\uac04\uc5d0\uc11c \uc911\uc2ec\uc774 \\(a\\)\uc778 \uac70\ub4ed\uc81c\uacf1\uae09\uc218 \\(\\sum a_n (x-a)^n\\)\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uc774 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc5b4\ub5bb\uac8c \uad6c\ud560\uae4c?<\/p>\n<p>\uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub294 \ub9c8\uce58 \uac70\ub300\ud55c \ub2e4\ud56d\uc2dd\ucc98\ub7fc \uc0dd\uacbc\ub2e4. \uc5b4\ub5a0\ud55c \uad00\uc810\uc5d0\uc11c\ub294 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub294 \ucc28\uc218\uac00 \ubb34\ud55c\ub300\uc778 \ub2e4\ud56d\uc2dd\ucc98\ub7fc \uc5ec\uae38 \uc218\ub3c4 \uc788\ub2e4. \uadf8\ub7f0\ub370 \ub2e4\ud56d\uc2dd\uc744 \uad6c\ud55c\ub2e4\ub294 \uac83\uc740 \ub2e4\ud56d\uc2dd\uc758 \uc0c1\uc218\uc640 \ubaa8\ub4e0 \ucc28\uc218\uc758 \ud56d\uc758 \uacc4\uc218\ub97c \uad6c\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \uac70\ub4ed\uc81c\uacf1\uae09\uc218 \\(\\sum a_n (x-a)^n\\)\uc744 \uad6c\ud55c\ub2e4\ub294 \uac83\uc740 \\(0\\) \uc774\uc0c1\uc758 \ubaa8\ub4e0 \uc815\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(a_k\\)\ub97c \uad6c\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4.<\/p>\n<p>\\(f\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \ud45c\ud604\ub41c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[f(x) = a_0 + a_1 (x-a)^1 + a_2 (x-a)^2 + a_3 (x-a)^3 + \\cdots + a_n (x-a)^n + \\cdots .\\tag{1.1}\\]<br \/>\n\uc774 \uc2dd\uc5d0 \\(x=a\\)\ub97c \ub300\uc785\ud558\uba74 \\(f(a) = a_0\\)\uc744 \uc5bb\ub294\ub2e4. (1.1)\uc758 \uc591\ubcc0\uc744 \ubbf8\ubd84\ud558\uba74<br \/>\n\\[f &#8216; (x) = a_1 + 2a_2 (x-a) + 3a_3 (x-a)^2 + \\cdots + na_n (x-a)^{n-1} + \\cdots\\tag{1.2}\\]<br \/>\n\uc774\uba70, \uc774 \uc2dd\uc5d0 \\(x=a\\)\ub97c \ub300\uc785\ud558\uba74 \\(f &#8216; (a) = a_1\\)\uc744 \uc5bb\ub294\ub2e4. (1.2)\uc758 \uc591\ubcc0\uc744 \ub2e4\uc2dc \ubbf8\ubd84\ud558\uba74<br \/>\n\\[f &#8216; &#8216; (x) = 2a_2 + 2\\cdot 3a_3 (x-a) + \\cdots + (n-1)n (x-a)^{n-2} + \\cdots\\tag{1.3}\\]<br \/>\n\uc774\uba70, \uc774 \uc2dd\uc5d0 \\(x=a\\)\ub97c \ub300\uc785\ud558\uba74 \\(f &#8216; &#8216; (a) = 2a_2\\) \uc989<br \/>\n\\[a_2 = \\frac{f &#8216; &#8216; (a)}{2}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4. \uc774 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec, (1.1)\uc744 \\(n\\)\ubc88 \ubbf8\ubd84\ud558\uba74<br \/>\n\\[f ^{(n)} (x) = n! a_n + (x-a)Q(x) \\tag{1.4}\\]<br \/>\n\uaf34\uc774 \ub418\uba70, \uc774 \uc2dd\uc5d0 \\(x=a\\)\ub97c \ub300\uc785\ud558\uba74 \\(f^{(n)}(a) = n! a_n\\) \uc989<br \/>\n\\[a_n = \\frac{f^{(n)}(a)}{n!} \\tag{1.5}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4. \ub9cc\uc57d \\(f^{(0)} = f\\)\ub77c\uace0 \ud55c\ub2e4\uba74 \\(0\\) \uc774\uc0c1\uc778 \ubaa8\ub4e0 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec (1.5)\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 1. (\ud14c\uc77c\ub7ec \uae09\uc218)<\/span><\/p>\n<p>\\(f\\)\uac00 \\(a\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc784\uc758 \ud69f\uc218\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \\(a\\)\uc5d0\uc11c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c <span class=\"defined\">\ud14c\uc77c\ub7ec \uae09\uc218<\/span>(Taylor series)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\sum_{n=1}^{\\infty} \\frac{f^{(n)}(a)}{n!} (x-a)^n.\\tag{1.6}\\]<br \/>\n\ud2b9\ud788 \uc911\uc2ec\uc774 \\(0\\)\uc778 \ud14c\uc77c\ub7ec \uae09\uc218\ub97c <span class=\"defined\">\ub9e5\ud074\ub77c\ub9b0 \uae09\uc218<\/span>(Maclaurin series)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n<p>\ubcf4\ud1b5 \ubcc4\ub2e4\ub978 \uc5b8\uae09 \uc5c6\uc774 \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uad6c\ud558\ub77c\uace0 \ud558\uba74 \uc911\uc2ec\uc774 \\(0\\)\uc778 \ud14c\uc77c\ub7ec \uae09\uc218, \uc989 \ub9e5\ud074\ub77c\ub9b0 \uae09\uc218\ub97c \uad6c\ud558\ub77c\ub294 \ub73b\uc73c\ub85c \ubc1b\uc544\ub4e4\uc774\uba74 \ub41c\ub2e4. \uc774\uc81c \uc6b0\ub9ac\uac00 \uc775\uc219\ud558\uac8c \uc0ac\uc6a9\ud588\ub358 \ud568\uc218\ub4e4\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uad6c\ud574 \ubcf4\uc790.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1. (\uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \ud14c\uc77c\ub7ec \uae09\uc218)<\/span><\/p>\n<p>\uc9c0\uc218\ud568\uc218 \\(f(x) = e^x\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uad6c\ud574 \ubcf4\uc790. \uba3c\uc800 \\(a_0 = f(0) = 1\\)\uc774\uba70, \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(f^{(n)}(x) = e^x\\)\uc774\ubbc0\ub85c<br \/>\n\\[a_n = \\frac{f^{(n)}(0)}{n!} = \\frac{e^0}{n!} = \\frac{1}{n!}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uad6c\ud558\ub294 \ud14c\uc77c\ub7ec \uae09\uc218\ub294<br \/>\n\\[e^x \\,\\sim\\, \\sum_{n=0}^{\\infty} \\frac{x^n}{n!} = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\frac{x^4}{4!} + \\cdots\\tag{1.7}\\]<br \/>\n\uc774\ub2e4. \uc774 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub294 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uc218\ub834\ud55c\ub2e4.<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p>\uc8fc\uc758\ud560 \uc810\uc740 \ubcf4\uae30 1\uc5d0\uc11c \uad6c\ud55c \ud14c\uc77c\ub7ec \uae09\uc218 (1.7)\uc774 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uc218\ub834\ud558\uae30\ub294 \ud558\uc9c0\ub9cc, \uadf8 \uadf9\ud55c\uc774 \\(e^x\\)\uc640 \uac19\ub2e4\ub294 \uac83\uc740 \ubcf4\uc7a5\ud560 \uc218 \uc5c6\ub2e4\ub294 \uac83\uc774\ub2e4. \uc989 \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \uc218\ub834\ud558\ub294 \uac83\uacfc \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \\(f\\)\uc758 \ud568\uc22b\uac12\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc740 \ub2e4\ub978 \ubb38\uc81c\uc774\ub2e4. \uc774\ub7ec\ud55c \uc774\uc720 \ub54c\ubb38\uc5d0 (1.7)\uc5d0\uc11c \ub4f1\ud638\ub97c \uc0ac\uc6a9\ud558\uc9c0 \uc54a\uace0 \ubb3c\uacb0\ud45c\uc2dc\ub97c \uc0ac\uc6a9\ud558\uc600\ub2e4.<\/p>\n<p>\ud14c\uc77c\ub7ec \uae09\uc218\uac00 \uc6d0\ub798\uc758 \ud568\uc218\uc5d0 \uc218\ub834\ud558\ub294\uc9c0 \uc5ec\ubd80\ub97c \ubc1d\ud788\ub294 \ubc29\ubc95\uc740 \uc774 \ud3ec\uc2a4\ud2b8\uc758 \ub4b7\ubd80\ubd84\uc5d0\uc11c \ud14c\uc77c\ub7ec \uc815\ub9ac\ub97c \ud1b5\ud574 \ubc1d\ud790 \uac83\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2. (\uc0ac\uc778\uc758 \ud14c\uc77c\ub7ec \uae09\uc218)<\/span><br \/>\n\uc0ac\uc778 \ud568\uc218 \\(f(x) = \\sin x\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uad6c\ud574 \ubcf4\uc790. \\(f\\)\uc758 \ub3c4\ud568\uc218\uc640 \\(0\\)\uc5d0\uc11c\uc758 \ubbf8\ubd84\uacc4\uc218\ub97c \uad6c\ud558\uba74<br \/>\n\\[\\begin{align}<br \/>\nf(x) &#038;= \\sin x &#038;\\quad f(0) &#038;= 0 \\\\[6pt]<br \/>\nf &#8216; (x) &#038;= \\cos x &#038;\\quad f &#8216; (0) &#038;= 1 \\\\[6pt]<br \/>\nf &#8216; &#8216; (x) &#038;= &#8211; \\sin x &#038;\\quad f &#8216; &#8216; (0) &#038;= 0 \\\\[6pt]<br \/>\nf^{(3)} (x) &#038;= -\\cos x &#038;\\quad f ^{(3)} (0) &#038;= -1 \\\\[6pt]<br \/>\nf^{(4)} (x) &#038;= \\sin x &#038;\\quad f^{(4)} (0) &#038;= 0 \\\\[6pt]<br \/>\nf^{(5)} (x) &#038;= \\cos x &#038;\\quad f^{(5)} (0) &#038;= 1 \\\\[6pt]<br \/>\n&#038;\\,\\,\\,\\vdots &#038;\\quad &#038;\\,\\,\\,\\vdots<br \/>\n\\end{align}\\]<br \/>\n\uc774\uba70, \uc774\uac83\uc740 \\(f^{(n+4)} = f^{(n)}\\)\uc73c\ub85c\uc11c \ubc18\ubcf5\ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\uc758 \uacc4\uc218\ub294<br \/>\n\\[\\begin{gather}<br \/>\na_0 = a_2 = a_4 = a_6 = \\cdots = 0, \\\\[6pt]<br \/>\n1! a_1 = 5! a_5 = 9! a_9 = 13! a_{13} = \\cdots = 1, \\\\[6pt]<br \/>\n3! a_3 = 7! a_7 = 11! a_{11} = 15! a_{15} = \\cdots = -1<br \/>\n\\end{gather}\\]<br \/>\n\uc774\uba70, \ud14c\uc77c\ub7ec \uae09\uc218\ub294<br \/>\n\\[\\sin x \\,\\sim\\, \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n+1}}{(2n+1)!} = x &#8211; \\frac{x^3}{3!} + \\frac{x^5}{5!} &#8211; \\frac{x^7}{7!} + &#8211; \\cdots \\tag{1.8}\\]<br \/>\n\uc774\ub2e4. \uc774 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub294 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uc218\ub834\ud55c\ub2e4.<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3. (\ucf54\uc0ac\uc778\uc758 \ud14c\uc77c\ub7ec \uae09\uc218)<\/span><br \/>\n\ucf54\uc0ac\uc778 \ud568\uc218 \\(f(x) = \\cos x\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uad6c\ud574 \ubcf4\uc790. \\(f\\)\uc758 \ub3c4\ud568\uc218\uc640 \\(0\\)\uc5d0\uc11c\uc758 \ubbf8\ubd84\uacc4\uc218\ub97c \uad6c\ud558\uba74<br \/>\n\\[\\begin{align}<br \/>\nf (x) &#038;= \\cos x &#038;\\quad f (0) &#038;= 1 \\\\[6pt]<br \/>\nf &#8216; (x) &#038;= &#8211; \\sin x &#038;\\quad f &#8216; (0) &#038;= 0 \\\\[6pt]<br \/>\nf^{(2)} (x) &#038;= -\\cos x &#038;\\quad f ^{(2)} (0) &#038;= -1 \\\\[6pt]<br \/>\nf^{(3)} (x) &#038;= \\sin x &#038;\\quad f^{(3)} (0) &#038;= 0 \\\\[6pt]<br \/>\nf^{(4)} (x) &#038;= \\cos x &#038;\\quad f^{(4)} (0) &#038;= 1 \\\\[6pt]<br \/>\nf^{(5)} (x) &#038;= -\\sin x &#038;\\quad f^{(5)} (0) &#038;= 0 \\\\[6pt]<br \/>\n&#038;\\,\\,\\,\\vdots &#038;\\quad &#038;\\,\\,\\,\\vdots<br \/>\n\\end{align}\\]<br \/>\n\uc774\uba70, \uc774\uac83\uc740 \\(f^{(n+4)} = f^{(n)}\\)\uc73c\ub85c\uc11c \ubc18\ubcf5\ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\uc758 \uacc4\uc218\ub294<br \/>\n\\[\\begin{gather}<br \/>\na_1 = a_3 = a_5 = a_7 = \\cdots = 0, \\\\[6pt]<br \/>\n2! a_2 = 6! a_6 = 10! a_{10} = 14! a_{14} = \\cdots = -1, \\\\[6pt]<br \/>\na_0 = 4! a_4 = 8! a_8 = 12! a_{12} = 16! a_{16} = \\cdots = 1<br \/>\n\\end{gather}\\]<br \/>\n\uc774\uba70, \ud14c\uc77c\ub7ec \uae09\uc218\ub294<br \/>\n\\[\\cos x \\,\\sim\\, \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n}}{(2n)!} = 1 &#8211; \\frac{x^2}{2!} + \\frac{x^4}{4!} &#8211; \\frac{x^6}{6!} + &#8211; \\cdots \\tag{1.9}\\]<br \/>\n\uc774\ub2e4. \uc774 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub294 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uc218\ub834\ud55c\ub2e4.<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4. (\uc790\uc5f0\ub85c\uadf8\uc758 \ud14c\uc77c\ub7ec \uae09\uc218)<\/span><br \/>\n\uc790\uc5f0\ub85c\uadf8 \ud568\uc218 \\(f(x) = \\ln (1+x)\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uad6c\ud574 \ubcf4\uc790. \\(f\\)\uc758 \ub3c4\ud568\uc218\uc640 \\(0\\)\uc5d0\uc11c\uc758 \ubbf8\ubd84\uacc4\uc218\ub97c \uad6c\ud558\uba74<br \/>\n\\[\\begin{align}<br \/>\nf (x) &#038;= \\ln (1+x) &#038;\\quad f (0) &#038;= 0 \\\\[6pt]<br \/>\nf &#8216; (x) &#038;= (1+x)^{-1} &#038;\\quad f &#8216; (0) &#038;= 1 \\\\[6pt]<br \/>\nf^{(2)} (x) &#038;= -(1+x)^{-2} &#038;\\quad f ^{(2)} (0) &#038;= -1 \\\\[6pt]<br \/>\nf^{(3)} (x) &#038;= 2(1+x)^{-3} &#038;\\quad f^{(3)} (0) &#038;= 2 \\\\[6pt]<br \/>\nf^{(4)} (x) &#038;= &#8211; 3! (1+x)^{-4} &#038;\\quad f^{(4)} (0) &#038;= -3! \\\\[6pt]<br \/>\nf^{(5)} (x) &#038;= 4! (1+x)^{-5} &#038;\\quad f^{(5)} (0) &#038;= 4! \\\\[6pt]<br \/>\n&#038;\\,\\,\\,\\vdots &#038;\\quad &#038;\\,\\,\\,\\vdots<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub294<br \/>\n\\[\\ln(1+x) \\,\\sim\\, \\sum_{n=1}^{\\infty} \\frac{(-1)^{n-1} x^n}{n} = x- \\frac{x^2}{2} + \\frac{x^3}{3} &#8211; \\frac{x^4}{4} + &#8211; \\cdots\\tag{1.10}\\]<br \/>\n\uc774\ub2e4. \uc774 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub294 \\(-1 < x \\le 1\\)\uc77c \ub54c \uc218\ub834\ud55c\ub2e4.<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"uniqueness\"><\/a><\/p>\n<h3>\ud14c\uc77c\ub7ec \uae09\uc218\uc758 \uc720\uc77c\uc131<\/h3>\n<p>\\(a=2\\)\uc5d0\uc11c \ubd84\uc218\ud568\uc218 \\(f(x) = 1\/x\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uad6c\ud574\ubcf4\uc790.<br \/>\n\\[f(x) = x^{-1} ,\\,\\, f &#8216; (x) = -x^{-2} ,\\,\\, f &#8216; &#8216; (x) = 2! x^{-3} ,\\,\\, \\cdots\\]<br \/>\n\uc774\uace0 \uc77c\ubc18\uc801\uc73c\ub85c<br \/>\n\\[f^{(n)} (x) = (-1)^n n! x^{-(n+1)}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\begin{align}<br \/>\na_0 &#038;= f(2) = \\frac{1}{2} ,\\\\[4pt]<br \/>\na_1 &#038;= f &#8216; (2) = &#8211; \\frac{1}{2^2} ,\\\\[4pt]<br \/>\na_2 &#038;= \\frac{f &#8216; &#8216; (2)}{2!} = \\frac{1}{2^3} , \\\\[4pt]<br \/>\n&#038;\\,\\,\\,\\vdots\\\\[4pt]<br \/>\na_n &#038;= \\frac{f^{(n)}(2)}{n!} = \\frac{(-1)^n}{2^{n+1}}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(2\\)\uc5d0\uc11c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub294<br \/>\n\\[\\frac{1}{2} &#8211; \\frac{x-2}{2^2} + \\frac{(x-2)^2}{2^3} &#8211; \\frac{(x-2)^3}{2^4} + &#8211; \\cdots\\tag{2.1}\\]<br \/>\n\uc774\ub2e4. \uc774 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub294 \\(0 < x < 4\\)\uc77c \ub54c \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\uc774\ubc88\uc5d0\ub294 \ub2e4\ub978 \ubc29\ubc95\uc73c\ub85c \\(2\\)\uc5d0\uc11c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uad6c\ud574\ubcf4\uc790. \ubd84\uc218\uc2dd \\(1\/x\\)\uc744 \ubcc0\ud615\ud558\uace0 \ubb34\ud55c\ub4f1\ube44\uae09\uc218(\uae30\ud558\uae09\uc218)\uc758 \ud569 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud558\uba74 \\(\\lvert x-2 \\rvert < 2\\)\uc77c \ub54c\n\\[\\begin{align}\n\\frac{1}{x} &#038;= \\frac{1}{2+(x-2)} \\\\[4pt]\n&#038;= \\frac{1}{2} \\cdot \\frac{1}{1 + \\frac{x-2}{2}} \\\\[2pt]\n&#038;= \\frac{1}{2} \\left\\{ 1 - \\frac{x-2}{2} + \\frac{(x-2)^2}{2^2} - \\frac{(x-2)^3}{2^3} + - \\cdots  \\right\\} \\\\[2pt]\n&#038;= \\frac{1}{2} - \\frac{x-2}{2^2} + \\frac{(x-2)^2}{2^3} - \\frac{(x-2)^3}{2^4} + - \\cdots \\tag{2.2}\n\\end{align}\\]\n\uc774\ubbc0\ub85c (2.1)\uc5d0\uc11c \uad6c\ud55c \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc640 \ub3d9\uc77c\ud558\ub2e4.<\/p>\n<p>\uadf8\ub807\ub2e4\uba74 \uc810 \\(a\\)\uc640 \ud568\uc218 \\(f\\)\uac00 \uace0\uc815\ub418\uc5b4 \uc788\uc744 \ub54c, \\(a\\)\uc5d0\uc11c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub418\ub294 \ud14c\uc77c\ub7ec \uae09\uc218\ub294 \uad6c\ud558\ub294 \ubc29\ubc95\uacfc\ub294 \uc0c1\uad00 \uc5c6\uc774 \ud56d\uc0c1 \ub3d9\uc77c\ud558\uac8c \ub098\uc62c\uae4c?<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 1. (\ud14c\uc77c\ub7ec \uae09\uc218\uc758 \uc720\uc77c\uc131)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uc810 \\(a\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \ud45c\ud604\ub41c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uadf8 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub294 \uc720\uc77c\ud558\uac8c \uacb0\uc815\ub41c\ub2e4. \uc989<br \/>\n\\[f(x) = \\sum_{n=0}^{\\infty} a_n (x-a)^n\\tag{2.3}\\]<br \/>\n\uc774\uace0<br \/>\n\\[f(x) = \\sum_{n=0}^{\\infty} b_n (x-a)^n\\tag{2.4}\\]<br \/>\n\uc774\uba74, \\(0\\) \uc774\uc0c1\uc778 \uc784\uc758\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n = b_n\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uba3c\uc800 (2.3)\uacfc (2.4)\uc5d0 \\(x=a\\)\ub97c \ub300\uc785\ud558\uba74 \\(a_0 = f(a) = b_0\\)\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(n\\)\uc774 \uc790\uc5f0\uc218\uc77c \ub54c, (2.3)\uc758 \uc591\ubcc0\uc744 \\(n\\)\ubc88 \ubbf8\ubd84\ud55c \ud6c4 \\(x=a\\)\ub97c \ub300\uc785\ud558\uba74<br \/>\n\\[f^{(n)}(a) = n! \\,a_n\\tag{2.5}\\]<br \/>\n\uc774\uba70, (2.4)\uc758 \uc591\ubcc0\uc744 \\(n\\)\ubc88 \ubbf8\ubd84\ud55c \ud6c4 \\(x=a\\)\ub97c \ub300\uc785\ud558\uba74<br \/>\n\\[f^{(n)}(a) = n! \\,b_n\\tag{2.6}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (2.5)\uc640 (2.6)\uc744 \uc5f0\ub9bd\ud558\uba74 \\(a_n = b_n\\)\uc744 \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uadf8\ub7ec\ubbc0\ub85c \uc8fc\uc5b4\uc9c4 \ud568\uc218\uc758 \uac70\ub4ed\uc81c\uacf1\uae09\uc218 \ud45c\ud604\uc744 \uad6c\ud560 \ub54c\uc5d0\ub294 \ubc18\ub4dc\uc2dc (1.6)\uc758 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud558\uc9c0 \uc54a\ub354\ub77c\ub3c4 \uc774\ubbf8 \uc54c\uace0 \uc788\ub294 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc774\uc6a9\ud558\uba74 \uac19\uc740 \uacb0\uacfc\ub97c \uc5bb\uc5b4\ub0bc \uc218 \uc788\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"convergence\"><\/a><\/p>\n<h3>\ud14c\uc77c\ub7ec \uae09\uc218\uc758 \uc218\ub834\uc131<\/h3>\n<p>\uc6b0\ub9ac\ub294 \ub450 \uac00\uc9c0 \uc9c8\ubb38 \uc911 \uccab \ubc88\uc9f8 \uc9c8\ubb38\uc758 \ub2f5\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc989 \ud568\uc218 \\(f\\)\uc758 \uac70\ub4ed\uc81c\uacf1\uae09\uc218 \ud45c\ud604\uc744 \uad6c\ud558\ub294 \ubc29\ubc95\uc73c\ub85c\uc11c \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774\uc81c \ub450 \ubc88\uc9f8 \uc9c8\ubb38\uc758 \ub2f5\uc744 \uc0b4\ud3b4\ubcf4\uc790. \uc989 \\(f\\)\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uad6c\ud588\uc744 \ub54c \uadf8 \uae09\uc218\uac00 \\(f\\)\uc5d0 \uc218\ub834\ud568\uc744 \uc99d\uba85\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 2. (\ud14c\uc77c\ub7ec\uc758 \uc815\ub9ac)<\/span><\/p>\n<p>\\(n\\)\uc774 \uc790\uc5f0\uc218\uc774\uace0 \\(a\\)\uc640 \\(x\\)\uac00 \uc11c\ub85c \ub2e4\ub978 \uc218\uc774\uba70, \\(I\\)\uac00 \\(a\\)\uc640 \\(x\\)\ub97c \ubaa8\ub450 \uc6d0\uc18c\ub85c \uac16\ub294 \uc5f4\ub9b0 \uad6c\uac04\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \ud568\uc218 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \\((n+1)\\)\ubc88 \uc774\uc0c1 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(a\\)\uc640 \\(x\\) \uc0ac\uc774\uc5d0 \uc810 \\(c\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[f(x) = f(a) + \\sum_{k=1}^{n} \\frac{f^{(k)}(a)}{k!} (x-a)^k + \\frac{f^{(n+1)}(c)}{(n+1)!} (x-a)^{n+1} \\tag{3.1}\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>(\ubbf8\uc801\ubd84\ud559\uc744 \ucc98\uc74c \uacf5\ubd80\ud558\ub294 \uc0ac\ub78c\uc740 \uc774 \uc99d\uba85\uc744 \uc0dd\ub7b5\ud574\ub3c4 \uc88b\ub2e4.)<\/p>\n<p>\\(a < x\\)\uc778 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(x > a\\)\uc778 \uacbd\uc6b0\uc758 \uc99d\uba85\ub3c4 \uc720\uc0ac\ud558\uae30 \ub54c\ubb38\uc774\ub2e4. \\(t\\in I\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\nF(t) &#038;:= \\frac{(x-t)^{n+1}}{(n+1)!} ,\\tag{3.2}\\\\[2pt]<br \/>\nG(t) &#038;:= f(x) &#8211; f(t) &#8211; \\sum_{k=1}^{n} \\frac{f^{(k)}(t)}{k!} (x-t)^k \\tag{3.3}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uc790. \uc774\uc81c \\(a\\)\uc640 \\(x\\) \uc0ac\uc774\uc5d0 \uc810 \\(c\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[G(a) = F(a) f^{(n+1)}(c) \\tag{3.4}\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0b4\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4. \ucf54\uc2dc\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uc790. \uac01 \\(t\\in I\\)\uc640 \uc790\uc5f0\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\frac{d}{dt} \\left( \\frac{f^{(k)}(t)}{k!} (x-t)^k \\right)<br \/>\n= \\frac{f^{(k+1)}(t)}{k!} (x-t)^k &#8211; \\frac{f^{(k)} (t)}{(k-1)!} (x-t)^{k-1}\\]<br \/>\n\uc774\ubbc0\ub85c \uc774 \uc2dd\uacfc (3.2)\ub97c \uacb0\ud569\ud558\uba74<br \/>\n\\[G &#8216; (t) = &#8211; \\frac{f^{(n+1)} (t)}{n!} (x-t)^n\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \ub610\ud55c (3.2)\uc758 \\(F\\)\ub97c \ubbf8\ubd84\ud558\uba74<br \/>\n\\[F &#8216; (t) = &#8211; \\frac{(x-t)^n}{n!} ,\\,\\, t\\in I\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4. \uc774\ub85c\uc368 \\(F\\)\uc640 \\(G\\)\ub294 \uc5f4\ub9b0 \uad6c\uac04 \\((a,\\,x)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \ub2eb\ud78c \uad6c\uac04 \\([a,\\,x]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba70, \\(t\\ne x\\)\uc77c \ub54c<br \/>\n\\[\\frac{G &#8216; (t)}{F &#8216; (t)} = f^{(n+1)}(t) \\tag{3.5}\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ub530\ub77c\uc11c \ucf54\uc2dc\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(a\\)\uc640 \\(x\\) \uc0ac\uc774\uc5d0 \uc810 \\(c\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[(F(x) &#8211; F(a)) G &#8216; (c) = (G(x) &#8211; G(a)) F &#8216; (C) \\tag{3.6}\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(F(x) = G(x) =0\\)\uc774\uace0 \\(x\\ne c\\)\uc774\ubbc0\ub85c<br \/>\n\\[-F(a) G &#8216; (c) = -F (a) F &#8216; (c)\\]<br \/>\n\uc989<br \/>\n\\[G(a) = F(a) \\frac{G &#8216; (c)}{F &#8216; (c)}\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774 \uc2dd\uc744 (3.5)\uc640 \uacb0\ud569\ud558\uba74 (3.4)\ub97c \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"remark\">\n<p><span class=\"remark\">\ucc38\uace0.<\/span><br \/>\n\uc815\ub9ac 2\uc5d0\uc11c \\(c\\)\ub294 \\(n\\)\uc758 \uac12\uc5d0 \ub530\ub77c \ubcc0\ud558\ub294 \uac12\uc774\ub2e4. \uc774 \ub54c\ubb38\uc5d0 \ucc45\uc5d0 \ub530\ub77c\uc11c\ub294 \uc815\ub9ac 2\ub97c \uae30\uc220\ud560 \ub54c \\(c\\)\ub97c \\(c_n\\)\uc73c\ub85c \ud45c\uc2dc\ud558\uae30\ub3c4 \ud55c\ub2e4.<br \/>\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p>\ud568\uc218 \\(f\\)\uac00 \\(a\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04 \\(I = (a-R ,\\, a+R)\\)\uc5d0\uc11c \uc784\uc758 \ud69f\uc218\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(a\\)\uc5d0\uc11c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \\(n\\)\ucc28 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc744 \\(P_n (x)\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c<br \/>\n\\[R_n (x) = f(x) &#8211; P_n (x) ,\\,\\, x\\in I\\]<br \/>\n\ub85c \uc815\uc758\ub41c \uc2dd \\(R_n\\)\uc744 \\(f\\)\uc758 \\(n\\)\ucc28 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc758 <span class=\"defined\">\ub098\uba38\uc9c0\uc2dd<\/span>(remainder)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9cc\uc57d \uc784\uc758\uc758 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\lim _ {n\\to\\infty} R_n (x) =0\\tag{3.7}\\]<br \/>\n\uc774\ub77c\uba74<br \/>\n\\[\\lim _ {n\\to\\infty} (f(x) &#8211; P_n (x)) =0\\]<br \/>\n\uc774\ubbc0\ub85c, \\(a\\)\uc5d0\uc11c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub294 \\(I\\)\uc5d0\uc11c \\(f\\)\uc640 \uc77c\uce58\ud55c\ub2e4. \uc989 \uc784\uc758\uc758 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec (3.7)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc740 \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \\(I\\)\uc5d0\uc11c \\(f\\)\uc640 \uc77c\uce58\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc774\ub2e4. \uadf8\ub7f0\ub370 \uc815\ub9ac 2\ub97c \uc774\uc6a9\ud558\uba74 \\(x \\ne a\\)\uc77c \ub54c \\(a\\)\uc640 \\(x\\) \uc0ac\uc774\uc5d0 \uc810 \\(c\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[R_n (x) = \\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}\\tag{3.8}\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc774 \uc2dd\uc744 \ud65c\uc6a9\ud558\ub294 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5. (\uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uc758 \uc218\ub834\uc131)<\/span><\/p>\n<p>\uc790\uc5f0\uc9c0\uc218\ud568\uc218 \\(f(x) = e^x\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218 (1.7)\uc774 \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc5d0 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc790. \\(x=0\\)\uc778 \uacbd\uc6b0 (1.7)\uc758 \uc88c\ubcc0\uc640 \uc6b0\ubcc0\uc774 \uc77c\uce58\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x \\ne 0\\)\uc778 \uacbd\uc6b0\ub9cc \uc0b4\ud3b4\ubcf4\uba74 \ub41c\ub2e4. \\(f\\)\uc758 \\(n\\)\ucc28 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc758 \ub098\uba38\uc9c0\ud56d\uc744 \\(R_n (x)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ud14c\uc77c\ub7ec\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[R_n (x) = \\frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}\\]<br \/>\n\uc778 \uc810 \\(c\\)\uac00 \\(0\\)\uacfc \\(x\\) \uc0ac\uc774\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[\\lvert f^{(n+1)}(c) \\rvert = \\lvert e^c \\rvert \\le e^{\\lvert c \\rvert} \\le e^{\\lvert x \\rvert}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\lvert R_n (x) \\rvert \\le \\frac{e^{\\lvert x \\rvert} \\lvert x \\rvert ^{n+1}}{(n+1)!}\\]<br \/>\n\uc774\ub2e4. \ubd80\ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc5d0 \uadf9\ud55c\uc744 \ucde8\ud558\uba74<br \/>\n\\[\\lim_{n\\to\\infty} \\frac{e^{\\lvert x \\rvert} \\lvert x \\rvert ^{n+1}}{(n+1)!} = 0\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\lim_{n\\to\\infty} R_n (x) =0\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(x\\)\ub294 \\(0\\)\uc774 \uc544\ub2cc \uc784\uc758\uc758 \uc2e4\uc218\uc774\ubbc0\ub85c, \uacb0\uad6d \ubaa8\ub4e0 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub294 \\(f\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc989 \ubaa8\ub4e0 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[e^x = \\sum_{n=0}^{\\infty} \\frac{x^n}{n!} = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\frac{x^4}{4!} + \\cdots\\tag{3.9}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p><span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6. (\uc0ac\uc778 \ud568\uc218\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uc758 \uc218\ub834\uc131)<\/span><\/p>\n<p><p>\uc0ac\uc778 \ud568\uc218 \\(f(x) = \\sin x\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218 (1.8)\uc774 \uc0ac\uc778 \ud568\uc218\uc5d0 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc790. \\(x=0\\)\uc778 \uacbd\uc6b0 (1.8)\uc758 \uc88c\ubcc0\uacfc \uc6b0\ubcc0\uc774 \uc77c\uce58\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x\\ne 0\\)\uc778 \uacbd\uc6b0\ub9cc \uc0b4\ud3b4\ubcf4\uba74 \ub41c\ub2e4. \\(f\\)\uc758 \\((2n+1)\\)\ucc28 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc758 \ub098\uba38\uc9c0\ud56d\uc744 \\(R_{2n+1} (x)\\)\ub77c\uace0 \ud558\uc790. (1.8)\uc758 \uc6b0\ubcc0\uc5d0 \ucc28\uc218\uac00 \ud640\uc218\uc778 \ud56d\ub9cc \uc874\uc7ac\ud558\ubbc0\ub85c \\(R_n\\) \ub300\uc2e0 \\(R_{2n+1}\\)\uc744 \uc0ac\uc6a9\ud55c\ub2e4. \uadf8\ub7ec\uba74 \ud14c\uc77c\ub7ec\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[R_{2n+1} (x) = \\frac{f^{(2n+2)}(c)}{(2n+2)!} x^{2n+2}\\]<br \/>\n\uc778 \uc810 \\(c\\)\uac00 \\(0\\)\uacfc \\(x\\) \uc0ac\uc774\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7f0\ub370 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\sin x\\)\uc758 \\(n\\)\uacc4\ub3c4\ud568\uc218\uc758 \uc808\ub313\uac12\uc740 \\(1\\)\uc744 \ub118\uc9c0 \uc54a\uc73c\ubbc0\ub85c<br \/>\n\\[\\lvert R_{2n+1} (x) \\rvert \\le \\frac{1}{ (2n+2)! } \\lvert x \\rvert^{2n+2}\\]<br \/>\n\uc774\ub2e4. \ubd80\ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc5d0 \uadf9\ud55c\uc744 \ucde8\ud558\uba74<br \/>\n\\[\\lim_{n\\to\\infty} \\frac{1}{ (2n+2)! } \\lvert x \\rvert^{2n+2} = 0\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\lim_{n\\to\\infty} R_{2n+1} (x) =0\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(x\\)\ub294 \\(0\\)\uc774 \uc544\ub2cc \uc784\uc758\uc758 \uc2e4\uc218\uc774\ubbc0\ub85c, \uacb0\uad6d \ubaa8\ub4e0 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub294 \\(f\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc989 \ubaa8\ub4e0 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\sin x = \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n+1}}{(2n+1)!} = x &#8211; \\frac{x^3}{3!} + \\frac{x^5}{5!} &#8211; \\frac{x^7}{7!} + &#8211; \\cdots \\tag{3.10}\\]<br \/>\n\uc774\ub2e4.<br \/>\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 7. (\ucf54\uc0ac\uc778 \ud568\uc218\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uc758 \uc218\ub834\uc131)<\/span><\/p>\n<p><p>\uc0ac\uc778 \ud568\uc218 \\(f(x) = \\cos x\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218 (1.9)\uac00 \ucf54\uc0ac\uc778 \ud568\uc218\uc5d0 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc790. \\(x=0\\)\uc778 \uacbd\uc6b0 (1.9)\uc758 \uc88c\ubcc0\uacfc \uc6b0\ubcc0\uc774 \uc77c\uce58\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x\\ne 0\\)\uc778 \uacbd\uc6b0\ub9cc \uc0b4\ud3b4\ubcf4\uba74 \ub41c\ub2e4. \\(f\\)\uc758 \\(2n\\)\ucc28 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc758 \ub098\uba38\uc9c0\ud56d\uc744 \\(R_{2n} (x)\\)\ub77c\uace0 \ud558\uc790. (1.9)\uc758 \uc6b0\ubcc0\uc5d0 \ucc28\uc218\uac00 \uc9dd\uc218\uc778 \ud56d\ub9cc \uc874\uc7ac\ud558\ubbc0\ub85c \\(R_n\\) \ub300\uc2e0 \\(R_{2n}\\)\uc744 \uc0ac\uc6a9\ud55c\ub2e4. \uadf8\ub7ec\uba74 \ud14c\uc77c\ub7ec\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[R_{2n} (x) = \\frac{f^{(2n+1)}(c)}{(2n+1)!} x^{2n+1}\\]<br \/>\n\uc778 \uc810 \\(c\\)\uac00 \\(0\\)\uacfc \\(x\\) \uc0ac\uc774\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7f0\ub370 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\cos x\\)\uc758 \\(n\\)\uacc4\ub3c4\ud568\uc218\uc758 \uc808\ub313\uac12\uc740 \\(1\\)\uc744 \ub118\uc9c0 \uc54a\uc73c\ubbc0\ub85c<br \/>\n\\[\\lvert R_{2n} (x) \\rvert \\le \\frac{1}{ (2n+1)! } \\lvert x \\rvert^{2n+1}\\]<br \/>\n\uc774\ub2e4. \ubd80\ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc5d0 \uadf9\ud55c\uc744 \ucde8\ud558\uba74<br \/>\n\\[\\lim_{n\\to\\infty} \\frac{1}{ (2n+1)! } \\lvert x \\rvert^{2n+1} = 0\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\lim_{n\\to\\infty} R_{2n} (x) =0\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(x\\)\ub294 \\(0\\)\uc774 \uc544\ub2cc \uc784\uc758\uc758 \uc2e4\uc218\uc774\ubbc0\ub85c, \uacb0\uad6d \ubaa8\ub4e0 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub294 \\(f\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc989 \ubaa8\ub4e0 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\cos x = \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n}}{(2n)!} = 1 &#8211; \\frac{x^2}{2!} + \\frac{x^4}{4!} &#8211; \\frac{x^6}{6!} + &#8211; \\cdots \\tag{3.11}\\]<br \/>\n\uc774\ub2e4.<br \/>\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 8. (\uc790\uc5f0\ub85c\uadf8\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uc758 \uc218\ub834\uc131)<\/span><\/p>\n<p>\uc790\uc5f0\ub85c\uadf8 \ud568\uc218 \\(f(x) = \\ln (1+x)\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218 (1.10)\uc774 \\(0 \\le x \\le 1\\)\uc77c \ub54c \uc790\uc5f0\ub85c\uadf8 \ud568\uc218\uc5d0 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc790. \\(x=0\\)\uc778 \uacbd\uc6b0 (1.10)\uc758 \uc88c\ubcc0\uacfc \uc6b0\ubcc0\uc774 \uc77c\uce58\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x > 0\\)\uc778 \uacbd\uc6b0\ub9cc \uc0b4\ud3b4\ubcf4\uba74 \ub41c\ub2e4. \\(f\\)\uc758 \\(n\\)\ucc28 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc758 \ub098\uba38\uc9c0\ud56d\uc744 \\(R_{n} (x)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ud14c\uc77c\ub7ec\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[R_{n} (x) = \\frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}\\]<br \/>\n\uc778 \uc810 \\(c\\)\uac00 \\(0\\)\uacfc \\(x\\) \uc0ac\uc774\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(f^{(n+1)} (c) = n! \/ (1+c)^{n+1}\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\lvert R_{n} (x) \\rvert = \\frac{\\lvert x \\rvert^{n+1}}{(n+1)(1+c)^{n+1}} = \\frac{1}{n+1} \\cdot \\left\\lvert \\frac{x}{1+c} \\right\\rvert ^{n+1}\\tag{3.12}\\]<br \/>\n\uc774\ub2e4. \\(\\lvert x \/ (1+c) \\rvert \\le 1\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\lim_{n\\to\\infty} \\frac{1}{n+1} \\left\\lvert \\frac{x}{1+c} \\right\\rvert = 0\\tag{3.13}\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[\\lim_{n\\to\\infty} R_n (x) =0\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(x\\)\ub294 \\(0\\)\ubcf4\ub2e4 \ud070 \uc784\uc758\uc758 \uc2e4\uc218\uc774\ubbc0\ub85c, \uacb0\uad6d \\(0 \\le x \\le 1\\)\uc778 \ubaa8\ub4e0 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub294 \\(f\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc989 \\(0 \\le x \\le 1\\)\uc77c \ub54c<br \/>\n\\[\\ln(1+x) = \\sum_{n=1}^{\\infty} \\frac{(-1)^{n-1} x^n}{n} = x- \\frac{x^2}{2} + \\frac{x^3}{3} &#8211; \\frac{x^4}{4} + &#8211; \\cdots\\tag{3.14}\\]<br \/>\n\uc774\ub2e4.<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p>\uc790\uc5f0\ub85c\uadf8\ud568\uc218\uc758 \ud14c\uc77c\ub7ec \uae09\uc218 (3.14)\ub294 \\(-1 < x \\le 1\\)\uc77c \ub54c \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ub098 \ubcf4\uae30 8\uc5d0\uc11c\ub294 \\(x \\ge 0\\)\uc77c \ub54c (3.14)\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \\(\\ln (1+x)\\)\uc5d0 \uc218\ub834\ud568\uc744 \ubcf4\uc600\ub2e4. \uadf8\ub807\ub2e4\uba74 \\(-1 < x < 0\\)\uc77c \ub54c\uc5d0\ub3c4 (3.14)\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \\(\\ln (1+x)\\)\uc5d0 \uc218\ub834\ud560\uae4c? \uc774\uac83\uc740 \uc815\ub9ac 2\ub97c \uc774\uc6a9\ud558\uc5ec \ubc1d\ud790 \uc218 \uc5c6\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(-1 < x < c < 0\\)\uc77c \ub54c (3.13)\uc758 \uadf9\ud55c\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\ub294 \uac83\uc744 \ubcf4\uc7a5\ud560 \uc218 \uc5c6\uae30 \ub54c\ubb38\uc774\ub2e4. \ub300\uc2e0 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc131\uc9c8\uc744 \uc774\uc6a9\ud558\uc5ec \uc774\uac83\uc744 \ubc1d\ud790 \uc218 \uc788\ub2e4. \\(\\lvert x \\rvert < 1\\)\uc77c \ub54c \ubb34\ud55c\ub4f1\ube44\uae09\uc218\uc758 \ud569 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud558\uba74\n\\[\\frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 + - \\cdots\\tag{3.15}\\]\n\uc774\ub2e4. \\(-1 < x < 1\\)\uc758 \ubc94\uc704\uc5d0\uc11c \uc704 \ub4f1\uc2dd\uc758 \uc591\ubcc0\uc758 \ubd80\uc815\uc801\ubd84\uc744 \uad6c\ud558\uba74\n\\[\\ln (1+x) = C+ x - \\frac{x^2}{2} + \\frac{x^3}{3} - \\frac{x^4}{4} + \\frac{x^5}{5} - \\frac{x^6}{6} + - \\cdots\\]\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(C\\)\ub294 \uc0c1\uc218\uc774\ub2e4. \uc774 \uc2dd\uc758 \uc591\ubcc0\uc5d0 \\(x=0\\)\uc744 \ub300\uc785\ud558\uba74 \\(C=0\\)\uc744 \uc5bb\ub294\ub2e4. \ub530\ub77c\uc11c\n\\[\\ln (1+x) =  x - \\frac{x^2}{2} + \\frac{x^3}{3} - \\frac{x^4}{4} + \\frac{x^5}{5} - \\frac{x^6}{6} + - \\tag{3.16}\\cdots\\]\n\uc774\ub2e4. \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \uc774 \ub4f1\uc2dd\uc740 \\(-1 < x < 1\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\uc774\ub7ec\ud55c \ubc29\ubc95\uc740 \ub85c\uadf8\ud568\uc218\uc758 \ub3c4\ud568\uc218\uac00 \uc77c\ucc28\ubd84\uc218\ud568\uc218\uc774\uace0 \uc77c\ucc28\ubd84\uc218\ud568\uc218\ub97c \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \uc27d\uac8c \ud45c\ud604\ud560 \uc218 \uc788\uae30 \ub54c\ubb38\uc5d0 \uac00\ub2a5\ud55c \uac83\uc774\ub2e4. \uc0ac\uc778, \ucf54\uc0ac\uc778, \uc9c0\uc218\ud568\uc218\uc758 \uacbd\uc6b0\uc5d0\ub294 \ubbf8\ubd84\ud558\uac70\ub098 \uc801\ubd84\ud558\uc600\uc744 \ub54c \uac70\ub4ed\uc81c\uacf1\uae09\uc218 \ud45c\ud604\uc774 \uc54c\ub824\uc9c4 \uac04\ub2e8\ud55c \ud568\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc5c6\uae30 \ub54c\ubb38\uc5d0 \uc774\ub7ec\ud55c \ubc29\ubc95\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc5c6\ub2e4.<\/p>\n<p>\ud55c\ud3b8 (3.16)\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\ub294 \uacfc\uc815\uc5d0\uc11c \\(x\\)\uc758 \ubc94\uc704\ub97c \\(-1 < x < 1\\)\uc774\ub77c\uace0 \ub450\uc5c8\ub294\ub370, \uadf8\uac83\uc740 (3.15)\uac00 \\(-1 < x < 1\\)\uc5d0\uc11c\ub9cc \uc218\ub834\ud558\uae30 \ub54c\ubb38\uc774\ub2e4. \\(x = 1\\)\uc77c \ub54c (3.16)\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\ub294 \uac83\uc740 \uc774\ub7ec\ud55c \ubc29\ubc95\uc73c\ub85c \ud560 \uc218 \uc5c6\ub2e4. \\(x = 1\\)\uc77c \ub54c (3.16)\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\ub294 \uac83\uc740 \ubcf4\uae30 8\uc758 \ubc29\ubc95\uc744 \uc774\uc6a9\ud574\uc57c \ud55c\ub2e4. \ud558\uc9c0\ub9cc \uc544\ubca8\uc758 \uc815\ub9ac(Abel's theorem)\ub97c \uc774\uc6a9\ud558\uba74 (3.16)\uc774 \uc131\ub9bd\ud558\ub294 \ubc94\uc704\ub97c \\(-1 < x \\le 1\\)\ub85c \uc27d\uac8c \ud655\uc7a5\ud560 \uc218 \uc788\ub2e4. \uc544\ubca8\uc758 \uc815\ub9ac\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 3. (\uc544\ubca8\uc758 \uc815\ub9ac)<\/span><\/p>\n<p>\uac70\ub4ed\uc81c\uacf1\uae09\uc218\uac00 \uc218\ub834\ud558\uba74 \uadf8 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub294 \uc218\ub834\uad6c\uac04 \ub0b4\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud568\uc218\uac00 \ub41c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p><a href=\"https:\/\/en.wikipedia.org\/wiki\/Abel%27s_theorem\">Wikipedia: Abel&#8217;s theorem<\/a>\uc744 \ucc38\uc870\ud558\ub77c. \uc77d\uc5b4\ub3c4 \uc774\ud574\uac00 \ub418\uc9c0 \uc54a\ub294\ub2e4\uba74 \uc99d\uba85\uc740 \uc0dd\ub7b5\ud558\uace0 \uc815\ub9ac\ub9cc \uc0ac\uc6a9\ud574\ub3c4 \ub41c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc544\ubca8\uc758 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uc5ec (3.16)\uc774 \\(x=1\\)\uc77c \ub54c\uc5d0\ub3c4 \uc131\ub9bd\ud568\uc744 \uc99d\uba85\ud574 \ubcf4\uc790. (3.16)\uc758 \uc6b0\ubcc0\uc758 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \\(T(x)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\ln(1+1) = T(1)\\)\uc784\uc744 \ubc1d\ud788\uba74 \ub41c\ub2e4. (3.16)\uc758 \uc6b0\ubcc0\uc758 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uac00 \\(-1 < x \\le 1\\)\uc77c \ub54c \uc218\ub834\ud558\ubbc0\ub85c \uc544\ubca8\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(T(x)\\)\ub294 \\(-1 < x \\le 1\\)\uc778 \ubc94\uc704\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \ub610\ud55c \\(\\ln(1+x)\\)\ub3c4 \\(-1 < x \\le 1\\)\uc778 \ubc94\uc704\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c\n\\[\\ln(1+1) = \\lim_{x\\to 1^-} \\ln (1+x) = \\lim_{x\\to 1^-} T(x) = T(1)\\]\n\uc774\ub2e4. \ub530\ub77c\uc11c (3.16)\uc740 \\(x=1\\)\uc77c \ub54c\uc5d0\ub3c4 \uc131\ub9bd\ud55c\ub2e4.\n\n\n<!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"approximation\"><\/a><\/p>\n<h3>\uc624\ucc28 \ucd94\uc815<\/h3>\n<p>\ud14c\uc77c\ub7ec\uc758 \uc815\ub9ac\ub294 \\(f\\)\uc758 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd \\(P_n\\)\uc774 \\(f\\)\uc5d0 \uc5bc\ub9c8\ub098 \uac00\uae4c\uc6b4\uc9c0, \uadf8 \uc624\ucc28\ub294 \uc5bc\ub9c8\ub098 \ub418\ub294\uc9c0 \ucd94\uc815\ud560 \ub54c \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \ub610\ud55c \uae30\uc900 \uc624\ucc28\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uc624\ucc28\uac00 \uae30\uc900 \uc624\ucc28\ub97c \ub118\uc9c0 \uc54a\ub3c4\ub85d \ud558\ub294 \\(x\\)\uc758 \ubc94\uc704\ub97c \uad6c\ud558\ub294 \ub370\uc5d0\ub3c4 \ud14c\uc77c\ub7ec\uc758 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 9. (\ud14c\uc77c\ub7ec \uc815\ub9ac\uc758 \ud65c\uc6a9)<\/span><\/p>\n<p>\\(x=0\\) \uadfc\ucc98\uc5d0\uc11c \\(\\sin x\\)\ub97c 3\ucc28 \ub2e4\ud56d\uc2dd \\(x &#8211; \\left( x^3 \/ 3! \\right)\\)\uc73c\ub85c \ub300\uccb4\ud558\ub824\uace0 \ud55c\ub2e4. \uc624\ucc28\uac00 \\(3\\times 10^{-4}\\)\uc744 \ub118\uc9c0 \uc54a\uac8c \uc720\uc9c0\ud558\ub824\uace0 \ud55c\ub2e4\uba74 \\(x\\)\ub294 \uc5b4\ub290 \ubc94\uc704\uc5d0 \uc788\uc5b4\uc57c \ud558\ub294\uc9c0 \uad6c\ud558\uc2dc\uc624.<\/p>\n<p><span class=\"proof\">\ud480\uc774.<\/span><br \/>\n\\(f(x) = \\sin x\\)\ub77c\uace0 \ud558\uba74 \ud14c\uc77c\ub7ec\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(x\\ne 0\\)\uc77c \ub54c<br \/>\n\\[\\left\\lvert \\sin x &#8211; \\left(x &#8211; \\frac{x^3}{3!} \\right) \\right\\rvert =  \\left\\lvert \\frac{f^{(4)}(c)}{4!} x^4 \\right\\rvert \\tag{4.1}\\]<br \/>\n\uc778 \uc810 \\(c\\)\uac00 \\(0\\)\uacfc \\(x\\) \uc0ac\uc774\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(\\left\\lvert f^{(4)}(c) \\right\\rvert \\le 1\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\left\\lvert \\frac{f^{(4)}(c)}{4!} x^4 \\right\\rvert \\le \\frac{\\lvert x \\rvert^4}{4!} \\tag{4.2} \\]<br \/>\n\uc774\ub2e4. \uc774 \uac12\uc774 \\(3 \\times 10^{-4}\\)\uc744 \ub118\uc9c0 \uc54a\uc544\uc57c \ud55c\ub2e4. \uc989 \ubd80\ub4f1\uc2dd<br \/>\n\\[\\frac{\\lvert x \\rvert^4}{4!} \\le \\frac{3}{10000}\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(x\\)\ub97c \uad6c\ud574\uc57c \ud55c\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc744 \ud480\uba74<br \/>\n\\[\\lvert x \\rvert \\le \\frac{\\sqrt[4]{72}}{10} \\approx 0.291\\tag{4.3}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p>\ud14c\uc77c\ub7ec\uc758 \uc815\ub9ac\ub294 \\(f\\)\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \\(f\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc54c\uace0 \uc788\uc9c0 \uc54a\uc744 \ub54c \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7f0\ub370 \uc6b0\ub9ac\ub294 \uc0ac\uc778 \ud568\uc218\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \uc0ac\uc778 \ud568\uc218\uc5d0 \uc218\ub834\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc54c\uace0 \uc788\ub2e4. \ub354\uc6b1\uc774<br \/>\n\\[\\sin x = x &#8211; \\frac{x^3}{3!} + \\frac{x^5}{5!} &#8211; \\frac{x^7}{7!} + &#8211; \\cdots \\tag{4.4}\\]<br \/>\n\uc774\ubbc0\ub85c \\(x\\ne 0\\)\uc77c \ub54c (4.4)\uc758 \uc6b0\ubcc0\uc740 \uad50\ub300\uae09\uc218\uac00 \ub41c\ub2e4. \ub530\ub77c\uc11c \uc0ac\uc778 \ud568\uc218\uc758 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc758 \uc624\ucc28\ub97c \uad6c\ud560 \ub54c\uc5d0\ub294 \uad50\ub300 \uae09\uc218\uc758 \uc624\ucc28 \ucd94\uc815 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud560 \uc218 \uc788\ub2e4. \uad50\ub300 \uae09\uc218 \uc624\ucc28 \ucd94\uc815 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec \ubcf4\uae30 9\ub97c \ud480\uc5b4\ubcf4\uc790.<br \/>\n\\[\\left\\lvert \\sin x &#8211; \\left(x &#8211; \\frac{x^3}{3!} \\right) \\right\\rvert \\le \\left\\lvert \\frac{x^5}{5!} \\right\\rvert \\tag{4.5}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\left\\lvert \\frac{x^5}{5!} \\right\\rvert \\le \\frac{3}{10000}\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(x\\)\ub97c \uad6c\ud574\uc57c \ud55c\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc744 \ud480\uba74<br \/>\n\\[\\lvert x \\rvert \\le \\sqrt[5]{0.036} \\approx 0.514\\tag{4.6}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4. \uc774 \ubc94\uc704\ub294 (4.3)\uc5d0\uc11c \uc5bb\uc740 \ubc94\uc704\ubcf4\ub2e4 \ub354 \ub113\ub2e4. \uc989 \ub3d9\uc77c\ud55c \uc870\uac74 \uc544\ub798\uc5d0\uc11c \uc0ac\uc6a9\ud560 \uc218 \uc788\ub294 \\(x\\)\uc758 \ubc94\uc704\uac00 \ub354 \ub113\ub2e4. \ub530\ub77c\uc11c (4.6)\uc774 (4.3)\ubcf4\ub2e4 \ub354 \uc88b\uc740 \uacb0\uacfc\uc774\ub2e4.<\/p>\n<p>(4.3)\ubcf4\ub2e4 (4.6)\uc5d0\uc11c \ub354 \uc88b\uc740 \uacb0\uacfc\ub97c \uc5bb\uc740 \uc774\uc720\ub294 \ub450 \uac00\uc9c0\uc774\ub2e4. \uccab\uc9f8, \uc0ac\uc778 \ud568\uc218\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \uc0ac\uc778 \ud568\uc218\uc5d0 \uc218\ub834\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc54c\uace0 \uc788\ub2e4. \ub458\uc9f8, \uc0ac\uc778 \ud568\uc218\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \uad50\ub300\uae09\uc218\uc774\ub2e4. \uc989 \ub354 \ub9ce\uc740 \uc815\ubcf4\ub97c \uac00\uc9c0\uace0 \uc2dc\uc791\ud560\uc218\ub85d \ub354 \uc88b\uc740 \uacb0\uacfc\ub97c \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc774\uac83\uc740 \uc624\uc9c0 \ud0d0\ud5d8\uc5d0 \ube44\uc720\ud560 \uc218 \uc788\ub2e4. \uc624\uc9c0\ub97c \ud0d0\ud5d8\ud558\uba74\uc11c \uc548\uc804\uc9c0\uc5ed(\\(x\\)\uc758 \ubc94\uc704)\uc744 \uc124\uc815\ud55c\ub2e4\uace0 \uc0dd\uac01\ud574\ubcf4\uc790. \uc5b4\ub5a0\ud55c \uc9c0\uc810\uc758 \uc815\ubcf4\ub97c \ucda9\ubd84\ud788 \uac16\uace0 \uc788\uc9c0 \uc54a\uc73c\uba74 \uadf8 \uc9c0\uc810\uc758 \uc548\uc815\uc131\uc744 \ubcf4\uc7a5\ud560 \uc218 \uc5c6\uc73c\ubbc0\ub85c \uadf8 \uc9c0\uc810\uc740 \uc548\uc804\uc9c0\uc5ed\uc5d0 \ud3ec\ud568\ub418\uc9c0 \uc54a\ub294\ub2e4. \ubc18\uba74 \uc5b4\ub5a0\ud55c \uc9c0\uc810\uc758 \uc815\ubcf4\ub97c \ucda9\ubd84\ud788 \uac16\uace0 \uc788\ub2e4\uba74 \uadf8 \uc9c0\uc810\uc744 \uc548\uc804\uc9c0\uc5ed\uc5d0 \ud3ec\ud568\uc2dc\ud0ac \uc218\ub3c4 \uc788\uace0 \uadf8\ub807\uc9c0 \uc54a\uc744 \uc218\ub3c4 \uc788\ub2e4. \uc989 \uc5b4\ub5a0\ud55c \uc9c0\uc810\uc758 \uc815\ubcf4\ub97c \ub354 \ub9ce\uc774 \uac00\uc9c0\uace0 \uc788\uc744\uc218\ub85d \uadf8 \uc9c0\uc810\uc740 \uc548\uc804\uc9c0\uc5ed\uc5d0 \ud3ec\ud568\ub420 \uac00\ub2a5\uc131\uc774 \ucee4\uc9c4\ub2e4. \uc694\ucee8\ub300 \ud574\ub2f9 \uc9c0\uc5ed\uc5d0 \ub300\ud55c \uc815\ubcf4\ub97c \ub354 \ub9ce\uc774 \uac00\uc9c0\uace0 \uc788\uc744\uc218\ub85d \uc548\uc804\ud558\ub2e4\uace0 \ud655\uc2e0\ud560 \uc218 \uc788\ub294 \uc9c0\uc5ed\uc758 \ubc94\uc704\ub294 \ub354 \ub113\uc5b4\uc9c4\ub2e4. (4.3)\uc5d0\uc11c \\(x\\)\uc758 \ubc94\uc704\ub97c \uad6c\ud560 \ub54c\ubcf4\ub2e4 (4.6)\uc5d0\uc11c \\(x\\)\uc758 \ubc94\uc704\ub97c \uad6c\ud560 \ub54c \uac00\uc9c0\uace0 \uc788\ub294 \uc815\ubcf4\uac00 \ub354 \ub9ce\uc558\ub2e4(\uc0ac\uc778 \ud568\uc218\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uc758 \uc218\ub834\uc131, \uad50\ub300\uae09\uc218\uc758 \uc131\uc9c8). \uc774\uac83\uc774 \ubc14\ub85c (4.6)\uc758 \ubc94\uc704\uac00 (4.3)\uc758 \ubc94\uc704\ubcf4\ub2e4 \ub354 \ub113\uc740 \uc774\uc720\uc774\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"nonanalyticfunctions\"><\/a><\/p>\n<h3>\ud14c\uc77c\ub7ec \uae09\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc5c6\ub294 \ud568\uc218\uc758 \uc608<\/h3>\n<p>\uc9c0\uae08\uae4c\uc9c0\uc758 \ub0b4\uc6a9\ub9cc \ubcf4\uba74 \ud568\uc218 \\(f\\)\uac00 \\(a\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc784\uc758 \ud69f\uc218\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c \\(a\\)\uc5d0\uc11c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub294 \ub2f9\uc5f0\ud788 \\(f\\)\ub85c \uc218\ub834\ud560 \uac83\ucc98\ub7fc \ubcf4\uc778\ub2e4. \ud558\uc9c0\ub9cc \uadf8\ub807\uc9c0 \uc54a\uc740 \ud568\uc218\uac00 \uc874\uc7ac\ud55c\ub2e4. \ud568\uc218 \\(f : \\mathbb{R} \\to \\mathbb{R}\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.<br \/>\n\\[f(x) = \\begin{cases}<br \/>\ne^{-1\/x^2} &#038; \\quad\\quad \\text{if} \\quad x \\ne 0 \\\\[6pt]<br \/>\n0 &#038; \\quad\\quad \\text{if} \\quad x = 0<br \/>\n\\end{cases}\\]<br \/>\n\uc774 \ud568\uc218\uac00 \ubaa8\ub4e0 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0, \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(f^{(n)} (0) = 0\\)\uc784\uc744 \ubcf4\uc774\uc790. \uba3c\uc800 \\(x\\ne 0\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uc6b0\uc120 \\(f\\)\uc758 \ub3c4\ud568\uc218\ub294<br \/>\n\\[f &#8216; (x) = \\frac{2}{x^3} e^{-1\/x^2}\\tag{5.1}\\]<br \/>\n\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(k\\)\uac00 \uc790\uc5f0\uc218\uc774\uace0 \ub450 \ub2e4\ud56d\uc2dd \\(P_k (x)\\)\uc640 \\(Q_k(x)\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[f ^{(k)} (x) = \\frac{P_k (x)}{Q_k (x)} e^{-1\/x^2}\\tag{5.2}\\]<br \/>\n\uc774<!-- \uba70, \\(P_k\\)\uc758 \ucc28\uc218\ubcf4\ub2e4 \\(Q_k\\)\uc758 \ucc28\uc218\uac00 \ub354 \ud06c\ub2e4 -->\ub77c\uace0 \uac00\uc815\ud558\uc790. (5.2)\ub97c \ubbf8\ubd84\ud558\uba74<br \/>\n\\[\\begin{align}<br \/>\nf^{(k+1)} (x)<br \/>\n&#038;= \\frac{P_k &#8216; (x) Q_k (x) &#8211; P_k (x) Q_k &#8216; (x)}{\\left\\{ Q_k (x) \\right\\}^2} e^{-1\/x^2} + \\frac{2 P_k (x)}{Q_k (x) x^3} e^{-1\/x^2} \\\\[3pt]<br \/>\n&#038;=  \\frac{\\left\\{P_k &#8216; (x) Q_k (x) &#8211; P_k (x) Q_k &#8216; (x)\\right\\} x^3 + 2 P_k (x) Q_k (x)}{x^3 \\left\\{  Q_k (x) \\right\\}^2} e^{-1\/x^2}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nP_{k+1} (x) &#038;= \\left\\{P_k &#8216; (x) Q_k (x) &#8211; P_k (x) Q_k &#8216; (x)\\right\\} x^3 + 2 P_k (x) Q_k (x) \\\\[6pt]<br \/>\nQ_{k+1} (x) &#038;= x^3 \\left\\{  Q_k (x) \\right\\}^2 \\tag{5.3}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uba74<br \/>\n\\[f^{(k+1)} (x) = \\frac{P_{k+1} (x)}{Q_{k+1} (x)} e^{-1\/x^2}\\]<br \/>\n\uc774<!-- \uba70, \\(P_{k+1}\\)\uc758 \ucc28\uc218\ubcf4\ub2e4 \\(Q_{k+1}\\)\uc758 \ucc28\uc218\uac00 \ub354 \ud06c -->\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \ub450 \ub2e4\ud56d\uc2dd \\(P_n (x)\\)\uc640 \\(Q_n (x)\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[f^{(n)} (x) = \\frac{P_{n} (x)}{Q_{n} (x)} e^{-1\/x^2}\\tag{5.4}\\]<br \/>\n\uc774<!-- \uba70, \\(P_n\\)\uc758 \ucc28\uc218\ubcf4\ub2e4 \\(Q_n\\)\uc758 \ucc28\uc218\uac00 \ub354 \ud06c -->\ub2e4. \ud2b9\ud788 (5.1), (5.2), (5.3)\uc744 \ubcf4\uba74<br \/>\n\\[Q_n (x) = x^{3\\times 2^n &#8211; 3}\\tag{5.5}\\]<br \/>\n\uc784\uc744 \uc54c \uc218 \uc788\ub2e4.<\/p>\n<p>\uc774\uc81c \\(0\\)\uc5d0\uc11c \\(f\\)\uc758 \\(n\\)\uacc4 \ubbf8\ubd84\uacc4\uc218\ub97c \uad6c\ud574\ubcf4\uc790. \uc6b0\uc120<br \/>\n\\[f &#8216; (0) = \\lim_{h\\to 0} \\frac{e^{-1\/h^2}}{h} = 0\\]<br \/>\n\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(k\\)\uac00 \uc790\uc5f0\uc218\uc774\uace0 \\(f^{(k)} (0) = 0\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. (5.5)\ub97c \uc774\uc6a9\ud558\uba74<br \/>\n\\[\\begin{align}<br \/>\nf^{(k+1)}(0)<br \/>\n&#038;= \\lim_{h\\to 0} \\frac{f^{(k)} (h) &#8211; f^{(k)} (0)}{h} \\\\[3pt]<br \/>\n&#038;= \\lim_{h\\to 0} \\frac{ P_k (h) e^{-1\/h^2}}{hQ_k (h)} \\\\[3pt]<br \/>\n&#038;= P_k (0) \\lim_{t\\to \\infty} \\frac{\\pm t}{Q_k (1\/t) e^{t^2}} \\\\[3pt]<br \/>\n&#038;= P_k (0) \\lim_{t\\to \\infty} \\frac{\\pm t \\cdot t^{3\\times 2^k -3}}{e^{t^2}} \\\\[3pt]<br \/>\n&#038;= P_k (0) \\times 0 = 0<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[f^{(n)}(0) = 0\\tag{5.6}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p><p>\\(f(0) = 0\\)\uc774\ub77c\ub294 \uc0ac\uc2e4\uacfc (5.6)\uc744 \uc774\uc6a9\ud558\uc5ec \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\uc758 \uc0c1\uc218\ud56d\uacfc \uacc4\uc218\ub97c \uad6c\ud558\uba74<br \/>\n\\[a_0 = a_1 = a_2 = \\cdots = 0\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub294<br \/>\n\\[T(x) = 0 + 0x + 0x^2 + 0x^3 + \\cdots = 0\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(x \\ne 0\\)\uc77c \ub54c \\(f(x) \\ne 0\\)\uc774\ubbc0\ub85c, \\(x\\ne 0\\)\uc77c \ub54c<br \/>\n\\[f(x) \\ne T(x)\\]<br \/>\n\uc774\ub2e4. \uc989 \\(f\\)\uac00 \uc784\uc758 \ud69f\uc218\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \uc874\uc7ac\ud558\uc9c0\ub9cc, \uadf8 \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \\(f\\)\ub97c \ub300\uccb4\ud560 \uc218 \uc5c6\ub2e4.\n<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### --><\/p>\n<h3>Meme<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=-BjZmE2gtdo\" target=\"_blank\" rel=\"noopener noreferrer\"><img fetchpriority=\"high\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2019\/06\/TaylorSeriesMeme.jpg\" alt=\"\" width=\"312\" height=\"321\" class=\"aligncenter size-full wp-image-4153\" style=\"border-style: solid; border-width: 4px; border-color: rgb(211,211,211);\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/06\/TaylorSeriesMeme.jpg 624w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/06\/TaylorSeriesMeme-292x300.jpg 292w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2019\/06\/TaylorSeriesMeme-585x602.jpg 585w\" sizes=\"(max-width: 312px) 100vw, 312px\" \/><\/a><\/p>\n<p><!--\n\n\n\n\n<div class=\"definition\">\n\n\n<p><span class=\"definition\">\uc815\uc758<\/span><\/p>\n\n\n\n\n<p> ... <\/p>\n\n<\/div>\n\n\n\n\n\n\n\n<div class=\"theorem\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac<\/span><\/p>\n\n\n\n\n<p> ... <\/p>\n\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85<\/p>\n\n\n\n\n<p>\n...\n<span class=\"qed\"><\/span><\/p>\n\n\n<\/div>\n\n\n\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 n.<\/span>\n...\n<span class=\"qee\"><\/span>\n<\/p>\n\n\n<\/div>\n\n\n\n\n\n\n<div class=\"remark\">\n\n\n<p><span class=\"remark\">\ucc38\uace0.<\/span>\n...\n<span class=\"qee\"><\/span>\n<\/p>\n\n\n<\/div>\n\n\n\n\n--><\/p>\n<p><!-- . --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uacf5\ubd80\ud560 \ub54c \uc911\uc810\uc801\uc73c\ub85c \uc0b4\ud3b4\ubcf4\uc544\uc57c \ud560 \ub0b4\uc6a9\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc138 \uac00\uc9c0\uc774\ub2e4. \uac70\ub4ed\uc81c\uacf1\uae09\uc218 \\(\\sum a_n (x-c)^n \\)\uc774 \uc218\ub834\ud558\ub3c4\ub85d \ud558\ub294 \\(x\\)\uc758 \uac12(\ubc94\uc704)\uc744 \uc5b4\ub5bb\uac8c \uad6c\ud560 \uac83\uc778\uac00? \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \uc815\uc758\ub41c \ud568\uc218\ub97c \ubbf8\ubd84\ud558\uac70\ub098 \uc801\ubd84\ud560 \ub550 \uc5b4\ub5bb\uac8c \ud558\ub294\uac00? \uc5b4\ub5a0\ud55c \ud568\uc218\ub97c \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub294\uac00? \uc774 \uc911 \uc138 \ubc88\uc9f8 \uc9c8\ubb38\uc5d0 \ub300\ud55c \ub2f5\uc774 \ubc14\ub85c \ud14c\uc77c\ub7ec \uae09\uc218\uc640 \ud14c\uc77c\ub7ec\uc758 \uc815\ub9ac\uc774\ub2e4. \ud14c\uc77c\ub7ec \uae09\uc218\ub294 \uc8fc\uc5b4\uc9c4 \ud568\uc218 \\(f\\)\ub97c \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \ub098\ud0c0\ub0b4\ub294 \ubc29\ubc95\uc744 \uc81c\uacf5\ud55c\ub2e4. \uc989 \ud568\uc218 \\(f\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c \uc774&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[],"class_list":["post-2024","post","type-post","status-publish","format-standard","hentry","category-calculus-ap"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2024","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"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=2024"}],"version-history":[{"count":99,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2024\/revisions"}],"predecessor-version":[{"id":4550,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2024\/revisions\/4550"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2024"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2024"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2024"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}