{"id":6775,"date":"2022-10-12T22:07:22","date_gmt":"2022-10-12T22:07:22","guid":{"rendered":"https:\/\/presswiki.allmath.gr\/wpwiki18\/?p=6775"},"modified":"2022-10-12T22:09:56","modified_gmt":"2022-10-12T22:09:56","slug":"%ce%b52001%ce%b34","status":"publish","type":"post","link":"https:\/\/presswiki.allmath.gr\/wpwiki18\/2022\/10\/12\/22\/07\/22\/6775\/","title":{"rendered":"\u03952001\u03b34"},"content":{"rendered":"\n<p>(\u03b1) \u039d\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03b5\u03c4\u03b5 \u03cc\u03c4\u03b9 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \u03b8\u03b5\u03c4\u03b9\u03ba\u03cc \u03b1\u03ba\u03ad\u03c1\u03b1\u03b9\u03bf n \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03cc\u03c4\u03b9:<\/p>\n\n\n\n<p>\\( \\left(\\frac{2n}{2n+1} \\right)^{2}&lt;\\frac{n}{n+1} \\)<\/p>\n\n\n\n<p>(\u03b2) \u039d\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03b5\u03c4\u03b5 \u03cc\u03c4\u03b9: <\/p>\n\n\n\n<p>\\( (\\frac{2}{3}.\\frac{4}{5}.\\frac{6}{7}. \u2026 .\\frac{2000}{2001})^{2}&lt;\\frac{1}{1001} \\)<\/p>\n\n\n\n<!--nextpage-->\n\n\n\n<figure class=\"wp-block-image size-full\"><img width=\"840\" height=\"320\" src=\"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-content\/uploads\/2022\/10\/\u03b5\u03b9\u03ba\u03cc\u03bd\u03b1-15.png\" alt=\"\" class=\"wp-image-6777\" srcset=\"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-content\/uploads\/2022\/10\/\u03b5\u03b9\u03ba\u03cc\u03bd\u03b1-15.png 840w, https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-content\/uploads\/2022\/10\/\u03b5\u03b9\u03ba\u03cc\u03bd\u03b1-15-300x114.png 300w, https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-content\/uploads\/2022\/10\/\u03b5\u03b9\u03ba\u03cc\u03bd\u03b1-15-768x293.png 768w\" sizes=\"(max-width: 709px) 85vw, (max-width: 909px) 67vw, (max-width: 1362px) 62vw, 840px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<p>\u03b1) \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \\(\\frac{4n^{2}}{4n^{2}+4n+1} &lt;\\frac{n}{n+1}\\)<br>\u0391\u03c0\u03bb\u03bf\u03c0\u03bf\u03b9\u03bf\u03cd\u03bc\u03b5 \u03c4\u03b1 \\( n \\) \u03c4\u03c9\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03b7\u03c4\u03ce\u03bd. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \\(n\\) \u03b8\u03b5\u03c4\u03b9\u03ba\u03cc\u03c2, \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03bd\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9<br>\\( 4n(n+1) &lt; 4n^{2}+4n+1 \\)<br>\\(4n^{2}+4n &lt; 4n^{2}+4n+1 \\)<br>\\(0&lt;1 \\), \u03c0\u03bf\u03c5 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9. \u03ac\u03c1\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03ba\u03b1\u03b9 \u03b7 \u03b1\u03c1\u03c7\u03b9\u03ba\u03ae \u03b1\u03bd\u03b9\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1.<br>\u03b2)<br>\\( (\\frac{2}{3})^{2}&lt;\\frac{1}{2} \\)<br>\\( (\\frac{4}{5})^{2}&lt;\\frac{2}{3} \\)<br>\\( (\\frac{6}{7})^{2}&lt;\\frac{3}{4} \\)<\/p>\n\n\n\n<p>\u03c3\u03c5\u03bd\u03b5\u03c7\u03af\u03b6\u03bf\u03bd\u03c4\u03b1\u03c2&#8230;<\/p>\n\n\n\n<p>\\( (\\frac{2000}{2001})^{2}&lt;\\frac{1000}{1001} \\)<\/p>\n\n\n\n<p>\u039f\u03c0\u03cc\u03c4\u03b5, \u03b1\u03c0\u03bb\u03bf\u03c0\u03bf\u03b9\u03ce\u03bd\u03c4\u03b1\u03c2, \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03b5\u03b9 \u03cc\u03c4\u03b9<br>\\( (\\frac{2}{3}.\\frac{4}{5}.\\frac{6}{7}. \u2026 .\\frac{2000}{2001})^{2}&lt;\\frac{1}{2}.\\frac{2}{3}.\\frac{3}{4}. \u2026 .\\frac{1000}{1001}=\\frac{1}{1001} \\)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>(\u03b1) \u039d\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03b5\u03c4\u03b5 \u03cc\u03c4\u03b9 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \u03b8\u03b5\u03c4\u03b9\u03ba\u03cc \u03b1\u03ba\u03ad\u03c1\u03b1\u03b9\u03bf n \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03cc\u03c4\u03b9: \\( \\left(\\frac{2n}{2n+1} \\right)^{2}&lt;\\frac{n}{n+1} \\) (\u03b2) \u039d\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03b5\u03c4\u03b5 \u03cc\u03c4\u03b9: \\( (\\frac{2}{3}.\\frac{4}{5}.\\frac{6}{7}. \u2026 .\\frac{2000}{2001})^{2}&lt;\\frac{1}{1001} \\)<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[238,17,228,54,62],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/posts\/6775"}],"collection":[{"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/comments?post=6775"}],"version-history":[{"count":2,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/posts\/6775\/revisions"}],"predecessor-version":[{"id":6779,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/posts\/6775\/revisions\/6779"}],"wp:attachment":[{"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/media?parent=6775"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/categories?post=6775"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/tags?post=6775"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}