\ubb34\ud55c\uae09\uc218 \uc218\ub834\ud310\uc815\uc744 \ud558\ub2e4 \ubcf4\uba74 \uc0bc\uac01\ud568\uc218\ub97c \ub2e4\ub8e8\uae30 \uc5b4\ub824\uc6b4 \uacbd\uc6b0\uac00 \uc788\ub2e4. \ub2e4\uc74c \ubb34\ud55c\uae09\uc218\ub97c \uc0b4\ud3b4\ubcf4\uc790.
\n\\[\\sum_{n=5}^{\\infty} \\left( \\frac{1}{n} – \\sin \\frac{n+1}{n^2 – 5n + 4} \\right ). \\tag{1}\\]
\n\\(n \\rightarrow 0\\)\uc77c \ub54c \uc0ac\uc778 \ud568\uc218 \uc548\uc5d0 \uc788\ub294 \ubd84\uc218\uc2dd\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\uace0, \\(x=0\\) \uadfc\ucc98\uc5d0\uc11c \\(\\sin x\\)\ub294 \\(x\\)\uc640 \ube44\uc2b7\ud558\uac8c \uc6c0\uc9c1\uc774\ubbc0\ub85c, \uc704 \ubb34\ud55c\uae09\uc218\uc5d0\uc11c \uc0ac\uc778\uc744 \uadf8\ub0e5 \uc5c6\uc560\uace0 \ud310\uc815\ud574\ub3c4 \ub420 \uac83 \uac19\ub2e4. \uc989 \uc704 \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub294
\n\\[\\sum_{n=5}^{\\infty} \\left( \\frac{1}{n} – \\frac{n+1}{n^2 -5n +4} \\right)\\]
\n\uc758 \uc218\ub834 \uc5ec\ubd80\uc640 \uac19\uc744 \uac83 \uac19\ub2e4. \uc774\ub807\uac8c \ub450\uace0 \ud310\uc815\ud574 \ubcf4\uba74 \uacb0\uacfc\uac00 \u2018\uc218\ub834\u2019\uc774 \ub098\uc624\uae30\ub294 \ud55c\ub2e4. \uadf8\ub7ec\ub098 \ub9c8\uc74c\ub300\ub85c \uc0ac\uc778\uc744 \uc5c6\uc560\uace0 \ud310\uc815\ud558\uba74 \uc548 \ub418\ub2c8, \uc774\ub7f0 \uacb0\uacfc\ub294 \ubbff\uc744 \uc218\uac00 \uc5c6\ub2e4.<\/p>\n
\uc6b0\ub9ac\ub294 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd \uadfc\uc0ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc0ac\uc778 \ud568\uc218\ub97c \ubcc0\ud615\ud558\ub294 \ubc29\ubc95\uc744 \ud1b5\ud574 \ubb34\ud55c\uae09\uc218 (1)\uc758 \uc218\ub834\uc744 \ud310\uc815\ud560 \uac83\uc774\ub2e4. \uc0ac\uc778 \ud568\uc218\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\sin x = x – \\frac{x^3}{3!} + \\frac{x^5}{5!} – \\frac{x^7}{7!} + – \\cdots .\\]
\n\uc774 \ub4f1\uc2dd\uc5d0\uc11c \uc6b0\ubcc0\uc758 \ubb34\ud55c\uae09\uc218\ub294 \uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uc88c\ubcc0\uc5d0 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c, \ub9cc\uc57d \\(x \\ge 0\\)\uc774\ub77c\uba74, \uad50\ub300\uae09\uc218\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc740 \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{aligned}
\n\\sin x & \\le x , \\\\[6pt]
\n\\sin x & \\ge x – \\frac{x^3}{3!} , \\\\[6pt]
\n\\sin x & \\le x – \\frac{x^3}{3!} + \\frac{x^5}{5!} , \\\\[6pt]
\n\\sin x & \\ge x – \\frac{x^3}{3!} + \\frac{x^5}{5!} – \\frac{x^7}{7!} , \\\\[6pt]
\n\\, & \\,\\, \\vdots
\n\\end{aligned}\\]
\n\uc989 \\(x \\ge 0\\)\uc77c \ub54c
\n\\[ x – \\frac{x^3}{3!} \\le \\sin x \\le x \\]
\n\uc774\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc5d0
\n\\[x = \\frac{n+1}{n^2 -5n +4}\\]
\n\ub97c \ub300\uc785\ud558\uba74, \\(n\\)\uc774 \\(n \\ge 5\\)\uc778 \uc790\uc5f0\uc218\uc77c \ub54c
\n\\[ \\frac{n+1}{n^2 -5n +4} – \\frac{n+1}{6\\left( n^2 -5n +4 \\right) } \\le \\sin \\left( \\frac{n+1}{n^2 -5n +4} \\right) \\le \\frac{n+1}{n^2 -5n +4} \\]
\n\uc989
\n\\[ \\frac{1}{n} – \\frac{n+1}{n^2 -5n +4} \\le \\frac{1}{n} –
\n\\sin \\left( \\frac{n+1}{n^2 -5n +4} \\right) \\le \\frac{1}{n} – \\frac{n+1}{n^2 -5n +4} + \\frac{1}{6} \\left\\{ \\frac{n+1}{\\left( n^2 -5n +4 \\right) }\\right\\}^3 \\]
\n\uc744 \uc5bb\ub294\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc758 \uac00\uc6b4\ub370\uc5d0 \uc788\ub294 \uc2dd\uc744 \\(a_n\\)\uc774\ub77c\uace0 \uc4f0\uace0, \ucc98\uc74c \uc2dd\uacfc \ub9c8\uc9c0\ub9c9 \uc2dd\uc744 \ubcc0\ud615\ud558\uba74
\n\\[ \\frac{(n^2 – 5n +4) -n(n+1)}{n( n^2 -5n +4 )} \\le a_n \\le \\frac{(n^2 – 5n +4) -n(n+1)}{n( n^2 -5n +4 )} + \\frac{1}{6} \\left\\{ \\frac{n+1}{\\left( n^2 -5n +4 \\right) }\\right\\}^3 \\]
\n\uc989
\n\\[ \\frac{ -6n +4 }{n^3 – 5n^2 + 4n} \\le a_n \\le \\frac{6(-6n+4)(n^2 – 5n+4)^2 + n(n+1)^3}{ 6n(n^2 -5n +4)^3} \\]
\n\uc744 \uc5bb\ub294\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc5d0\uc11c \uac00\uc7a5 \ub9c8\uc9c0\ub9c9 \uc2dd\uc744 \uc0b4\ud3b4\ubcf4\uc790. \uc774 \uc2dd\uc758 \ubd84\uc790\uc640 \ubd84\ubaa8\ub294 \uac01\uac01 \\(n\\)\uc5d0 \ub300\ud55c \ub2e4\ud56d\uc2dd\uc774\ub2e4. \ubd84\uc790\ub294 \ucd5c\uace0\ucc28\ud56d\uc774 \\(-36n^5\\)\uc774\uba70, \ubd84\ubaa8\ub294 \ucd5c\uace0\ucc28\ud56d\uc774 \\(6n^7\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(n\\)\uc758 \uac12\uc774 \ucda9\ubd84\ud788 \ud074 \ub54c, \ub9c8\uc9c0\ub9c9 \uc2dd\uc740 \uc74c\uc218\uc774\ub2e4. \uc989 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(n \\ge N\\)\uc778 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[ \\frac{ -6n +4 }{n^3 – 5n^2 + 4n} \\le a_n \\le \\frac{6(-6n+4)(n^2 – 5n+4)^2 + n(n+1)^3}{ 6n(n^2 -5n +4)^3} \\le 0\\]
\n\uc774\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc73c\ub85c\ubd80\ud130, \\(n \\ge N\\)\uc77c \ub54c
\n\\[\\frac{ -6n +4 }{n^3 – 5n^2 + 4n} \\le a_n \\le 0\\]
\n\uc989
\n\\[0 \\le -a_n \\le \\frac{ 6n -4 }{n^3 – 5n^2 + 4n} \\tag{2}\\]
\n\uc744 \uc5bb\ub294\ub2e4.
\n\\[b_n = \\frac{ 6n -4 }{n^3 – 5n^2 + 4n} ,\\quad c_n = \\frac{1}{n^2} \\]
\n\uc774\ub77c\uace0 \ud558\uba74 \\(n \\ge 5\\)\uc77c \ub54c \\(b_n \\ge 0 ,\\) \\(c_n > 0\\)\uc774\uace0
\n\\[\\lim_{n\\rightarrow\\infty} \\frac{b_n}{c_n} = 6\\]
\n\uc774\uba70, \\(\\sum_{n=N}^{\\infty} c_n\\)\uc774 \uc218\ub834\ud558\ubbc0\ub85c(\u2235 \\(p=2\\)\uc778 \\(p\\)\uae09\uc218), \uadf9\ud55c\ube44\uad50\ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(\\sum_{n=N}^{\\infty} b_n \\)\ub3c4 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ubd80\ub4f1\uc2dd (2)\uc640 \uc9c1\uc811\ube44\uad50\ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec
\n\\[\\sum_{n=N}^{\\infty} (-a_n)\\]
\n\ub3c4 \uc218\ub834\ud55c\ub2e4. \uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\uc758 \ubaa8\ub4e0 \ud56d\uc758 \ubd80\ud638\ub97c \ubc14\uafbc \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud558\uace0, \uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\uc5d0 \uc720\ud55c \uac1c\uc758 \ud56d\uc744 \ucd94\uac00\ud558\uac70\ub098 \uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\uc5d0\uc11c \uc720\ud55c \uac1c\uc758 \ud56d\uc744 \uc81c\uac70\ud55c \ubb34\ud55c\uae09\uc218\ub3c4 \uc218\ub834\ud558\ubbc0\ub85c
\n\\[\\sum_{n=5}^{\\infty} a_n\\]
\n\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/p>\n","protected":false},"excerpt":{"rendered":"
\ubb34\ud55c\uae09\uc218 \uc218\ub834\ud310\uc815\uc744 \ud558\ub2e4 \ubcf4\uba74 \uc0bc\uac01\ud568\uc218\ub97c \ub2e4\ub8e8\uae30 \uc5b4\ub824\uc6b4 \uacbd\uc6b0\uac00 \uc788\ub2e4. \ub2e4\uc74c \ubb34\ud55c\uae09\uc218\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(\\sum_{n=5}^{\\infty} \\left( \\frac{1}{n} – \\sin \\frac{n+1}{n^2 – 5n + 4} \\right ). \\) \\(n \\rightarrow 0\\)\uc77c \ub54c \uc0ac\uc778 \ud568\uc218 \uc548\uc5d0 \uc788\ub294 \ubd84\uc218\uc2dd\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\uace0, \\(x=0\\) \uadfc\ucc98\uc5d0\uc11c \\(\\sin x\\)\ub294 \\(x\\)\uc640 \ube44\uc2b7\ud558\uac8c \uc6c0\uc9c1\uc774\ubbc0\ub85c, \uc704 \ubb34\ud55c\uae09\uc218\uc5d0\uc11c \uc0ac\uc778\uc744 \uadf8\ub0e5 \uc5c6\uc560\uace0 \ud310\uc815\ud574\ub3c4 \ub420 \uac83 \uac19\ub2e4. \uc989 \uc704 \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub294 \\(\\sum_{n=5}^{\\infty} \\left( \\frac{1}{n} – \\frac{n+1}{n^2 -5n +4}…<\/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":[327,582,583,360],"class_list":["post-8879","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-327","tag-582","tag-583","tag-360"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8879","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=8879"}],"version-history":[{"count":39,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8879\/revisions"}],"predecessor-version":[{"id":8918,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8879\/revisions\/8918"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=8879"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=8879"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=8879"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}