\u4eca\u671d\uff0c\u3075\u3068\u601d\u3044\u3064\u3044\u3066\u8a08\u7b97\u3057\u3066\u307f\u305f\u3089\uff0c\u3042\u307e\u308a\u306b\u3082\u7c21\u5358\u306b\u3067\u304d\u3066\uff0c\u9a5a\u3044\u305f\u3002<\/p>\n
$n$\u3092\u4efb\u610f\u306e\u6b63\u306e\u6574\u6570\u3068\u3059\u308b\u3068\u304d\uff0c$x>0$\u306b\u304a\u3044\u3066\u4e0d\u7b49\u5f0f
\n\\[ e^x>1+x+\\frac{x^2}{2!}+\\frac{x^3}{3!}+\\cdots+\\frac{x^n}{n!} \\]
\n\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u306f\u3088\u304f\u77e5\u3089\u308c\u3066\u3044\u308b\u3002$e^x$\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u304b\u3089\u81ea\u7136\u306b\u5c0e\u304b\u308c\u308b\u306e\u3067\u3042\u308b\u304c\uff0c\u305d\u308c\u3092\u4f7f\u308f\u305a\u306b\u76f4\u63a5\u306b\u8a3c\u660e\u3057\u3088\u3046\u3068\u3059\u308b\u3068\uff0c\u5fae\u5206\u3092\u4f55\u56de\u3082\u3059\u308b\u3053\u3068\u306b\u306a\u308a(\u5e30\u7d0d\u6cd5\u3068\u306e\u4f75\u7528)\uff0c\u3051\u3063\u3053\u3046\u624b\u9593\u304c\u304b\u304b\u308b\u306e\u3067\u3042\u308b\u3002<\/p>\n
\u3068\u3053\u308d\u3067\u3042\u308b\u3002\u4e21\u8fba\u306e\u5bfe\u6570\u3092\u3068\u3063\u305f\u4e0d\u7b49\u5f0f
\n\\[ x>\\log\\biggl(1+x+\\frac{x^2}{2!}+\\frac{x^3}{3!}+\\cdots+\\frac{x^n}{n!}\\biggr) \\]
\n\u3092\u8003\u3048\u308b\u3068\uff0c\u3042\u3089\u4e0d\u601d\u8b70\u3002\u5fae\u5206\u4e00\u56de\u3060\u3051\u3067\u7c21\u5358\u306b\u8a3c\u660e\u3067\u304d\u308b\u306e\u3067\u3042\u308b\u3002
\n\\[ f_n(x)=1+x+\\frac{x^2}{2!}+\\frac{x^3}{3!}+\\cdots+\\frac{x^n}{n!} \\]
\n\u3068\u304a\u3051\u3070\uff0c
\n\\[ f_n'(x)=f_{n-1}(x), \\qquad f_{n}(x)=f_{n-1}(x)+\\frac{x^n}{n!} \\]
\n\u304c\u6210\u308a\u7acb\u3064\u304b\u3089\uff0c
\n\\[ g(x)=x-\\log f_n(x) \\]
\n\u3092\u5fae\u5206\u3059\u308b\u3068\uff0c
\n\\[ g'(x)=1-\\frac{f_n'(x)}{f_n(x)}=1-\\frac{f_{n-1}(x)}{f_n(x)}=\\frac{x^n}{n!\\,f_n(x)} \\]
\n\u3068\u306a\u308b\u3002\u3053\u308c\u304b\u3089$g(x)$\u306f$x\\geqq 0$\u306b\u304a\u3044\u3066\u5358\u8abf\u5897\u52a0\u3067\u3042\u308b\u3053\u3068\u304c\u5206\u304b\u308b\u304b\u3089\uff0c$x>0$\u306b\u304a\u3044\u3066$g(x)>0$\u3067\u3042\u308b\u3053\u3068\u304c\u8a00\u3048\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u4eca\u671d\uff0c\u3075\u3068\u601d\u3044\u3064\u3044\u3066\u8a08\u7b97\u3057\u3066\u307f\u305f\u3089\uff0c\u3042\u307e\u308a\u306b\u3082\u7c21\u5358\u306b\u3067\u304d\u3066\uff0c\u9a5a\u3044\u305f\u3002 $n$\u3092\u4efb\u610f\u306e\u6b63\u306e\u6574\u6570\u3068\u3059\u308b\u3068\u304d\uff0c$x>0$\u306b\u304a\u3044\u3066\u4e0d\u7b49\u5f0f \\[ e^x>1+x+\\frac{x^2}{2!}+\\frac{x^3}{3!}+\\cdo […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-2163","post","type-post","status-publish","format-standard","hentry","category-5"],"_links":{"self":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/2163","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/comments?post=2163"}],"version-history":[{"count":10,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/2163\/revisions"}],"predecessor-version":[{"id":4101,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/2163\/revisions\/4101"}],"wp:attachment":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/media?parent=2163"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/categories?post=2163"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/tags?post=2163"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}