{"id":6368,"date":"2021-05-07T06:00:59","date_gmt":"2021-05-07T04:00:59","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6368"},"modified":"2021-05-14T08:25:45","modified_gmt":"2021-05-14T06:25:45","slug":"suma-de-serie-racional","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/suma-de-serie-racional\/","title":{"rendered":"Suma de serie racional"},"content":{"rendered":"<p>El enunciado del problema 6 de la <a href=\"https:\/\/bit.ly\/3aIfOT4\">Fase Local de la Olimpiada Matem\u00e1tica Espa\u00f1ola del 2020<\/a> es<\/p>\n<blockquote><p>\n  Sea n un entero positivo. Calcular la siguiente suma\n<\/p><\/blockquote>\n<pre lang=\"text\">\n         3           4           5                    n+2\n     --------- + --------- + --------- + \u00b7\u00b7\u00b7 + ---------------------\n      1\u00b72\u00b74\u00b75     2\u00b73\u00b75\u00b76     3\u00b74\u00b76\u00b77           n\u00b7(n+1)\u00b7(n+3)\u00b7(n+4)\n<\/pre>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   sumaSerie :: Integer -> Rational\n<\/pre>\n<p>tal que para cada entero positivo n, (sumaSerie n) es el valor de la siguiente sumaSerie<\/p>\n<pre lang=\"text\">\n      3           4           5                    n+2\n  --------- + --------- + --------- + \u00b7\u00b7\u00b7 + ---------------------\n   1\u00b72\u00b74\u00b75     2\u00b73\u00b75\u00b76     3\u00b74\u00b76\u00b77           n\u00b7(n+1)\u00b7(n+3)\u00b7(n+4)\n<\/pre>\n<p>Por ejemplo,<\/p>\n<pre lang=\"text\">\n   sumaSerie 1        ==  3 % 40\n   sumaSerie 2        ==  7 % 72\n   sumaSerie 3        ==  3 % 28\n   sumaSerie (10^10)  ==  3125000001562500000 % 25000000012500000001\n\n   length (show (sumaSerie (10^10)))      ==  42\n   length (show (sumaSerie (10^(10^2))))  ==  402\n   length (show (sumaSerie (10^(10^3))))  ==  4002\n   length (show (sumaSerie (10^(10^4))))  ==  40002\n   length (show (sumaSerie (10^(10^5))))  ==  400002\n   length (show (sumaSerie (10^(10^6))))  ==  4000002\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck (Property, (==>), quickCheck)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nsumaSerie :: Integer -> Rational\nsumaSerie n =\n  sum [k\/((k-1)*(k-2)*(k+1)*(k+2))\n      | a <- [3..n+2],\n        let k = fromIntegral a]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\n-- Calculando los primeros t\u00e9rminos\n--    \u03bb> [sumaSerie n | n <- [1..11]]\n--    [3 % 40,7 % 72,3 % 28,9 % 80,25 % 216,33 % 280,21 % 176,13 % 108,63 % 520,75 % 616,11 % 90]\n-- y usando Wolfram Alpha https:\/\/bit.ly\/2PvoCEK\n\nsumaSerie2 :: Integer -> Rational\nsumaSerie2 n = m*(m+5)\/(8*(m+1)*(m+4))\n  where m = fromIntegral n\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_sumaSerie :: Integer -> Property\nprop_sumaSerie n =\n  n > 0 ==>\n  sumaSerie n == sumaSerie2 n\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_sumaSerie\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> sumaSerie (10^6)\n--    31250156250 % 250001250001\n--    (4.99 secs, 6,029,246,016 bytes)\n--    \u03bb> sumaSerie2 (10^6)\n--    31250156250 % 250001250001\n--    (0.02 secs, 123,280 bytes)\n<\/pre>\n<ul>\n<li>En los comentarios se pueden escribir nuevas soluciones.\n<li>El c\u00f3digo se debe escribir entre una l\u00ednea con &#60;pre lang=&quot;haskell&quot;&#62; y otra con &#60;\/pre&#62;\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>El enunciado del problema 6 de la Fase Local de la Olimpiada Matem\u00e1tica Espa\u00f1ola del 2020 es Sea n un entero positivo. Calcular la siguiente suma 3 4 5 n+2 &#8212;&#8212;&#8212; + &#8212;&#8212;&#8212; + &#8212;&#8212;&#8212; + \u00b7\u00b7\u00b7 + &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; 1\u00b72\u00b74\u00b75 2\u00b73\u00b75\u00b76 3\u00b74\u00b76\u00b77 n\u00b7(n+1)\u00b7(n+3)\u00b7(n+4) Definir la funci\u00f3n sumaSerie :: Integer -> Rational tal que para cada&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[2],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6368"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=6368"}],"version-history":[{"count":9,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6368\/revisions"}],"predecessor-version":[{"id":6461,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6368\/revisions\/6461"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6368"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6368"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6368"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}