{"id":3649,"date":"2013-09-20T18:29:36","date_gmt":"2013-09-20T16:29:36","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=3649"},"modified":"2013-09-21T18:39:33","modified_gmt":"2013-09-21T16:39:33","slug":"problema-de-ramanujan-de-radicales-anidados","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/problema-de-ramanujan-de-radicales-anidados\/","title":{"rendered":"Problema de Ramanujan de radicales anidados"},"content":{"rendered":"<p>El viernes pasado, John D. Cook escribi\u00f3 el art\u00edculo <a href=\"http:\/\/bit.ly\/16aLkh7\">Ramanujan\u2019s nested radical<\/a> en el que comenta el siguiente problema planteado por <a href=\"http:\/\/es.wikipedia.org\/wiki\/Srinivasa_Aiyangar_Ramanujan\">Ramanujan<\/a> en el <i>Journal of Indian Mathematical Society<\/i>: <\/p>\n<blockquote><p>\n\u00bfCu\u00e1l es el valor de la siguiente expresi\u00f3n?<br \/>\n<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Csqrt%7B1%2B2%5Csqrt%7B1%2B3%5Csqrt%7B1%2B4%5Csqrt%7B1%2B%5Cldots%7D%7D%7D%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;sqrt{1+2&#92;sqrt{1+3&#92;sqrt{1+4&#92;sqrt{1+&#92;ldots}}}}\" class=\"latex\" \/>\n<\/p><\/blockquote>\n<p>A partir del problema he elaborado la siguiente relaci\u00f3n de ejercicios en Haskell para la asignatura de <a href=\"http:\/\/www.cs.us.es\/~jalonso\/cursos\/i1m\">Inform\u00e1tica (de 1\u00ba del Grado en Matem\u00e1ticas)<\/a>.<br \/>\n<!--more--><\/p>\n<pre lang=\"haskell\">\r\n-- ---------------------------------------------------------------------\r\n-- Librer\u00edas auxiliares\r\n-- ---------------------------------------------------------------------\r\n \r\nimport Graphics.Gnuplot.Simple \r\n\r\n-- ---------------------------------------------------------------------\r\n-- Ejercicios\r\n-- ---------------------------------------------------------------------\r\n\r\n-- ---------------------------------------------------------------------\r\n-- Ejercicio 1. Definir la funci\u00f3n\r\n--    ramanujan :: Double -> Double\r\n-- tal que (ramanujan n) es el n-\u00e9simo t\u00e9rmino de la sucesi\u00f3n de\r\n-- Ramanujan correspondiente a la expresi\u00f3n\r\n--    sqrt(1+2*sqrt(1+3*sqrt(1+4*sqrt(1+...))))\r\n-- es decir,\r\n--    ramanujan(1) = 1\r\n--    ramanujan(2) = sqrt(1+2)\r\n--    ramanujan(3) = sqrt(1+2*sqrt(1+3))\r\n--    ramanujan(4) = sqrt(1+2*sqrt(1+3*sqrt(1+4)))\r\n-- Por ejemplo,\r\n--    ramanujan 4  ==  2.559830165300118\r\n-- ---------------------------------------------------------------------\r\n\r\n-- 1\u00aa soluci\u00f3n:\r\nramanujan :: Double -> Double\r\nramanujan n = f [2..n]\r\n  where f []     = 1\r\n        f (x:xs) = sqrt (1 + x * f xs)\r\n\r\n-- 2\u00aa soluci\u00f3n:\r\nramanujan2 :: Double -> Double\r\nramanujan2 n = aux n\r\n    where aux 1 = n\r\n          aux m = (n-m+1) * sqrt (1 + aux (m-1))\r\n\r\n-- ---------------------------------------------------------------------\r\n-- Ejercicio 2. Definir la funci\u00f3n\r\n--    limite :: (Num a, Enum a, Num b, Ord b) => (a -> b) -> b -> b\r\n-- tal que (limite f a) es el valor de f en el primer t\u00e9rmino x tal que \r\n-- para todo y entre x+1 y x+100, el valor absoluto de f(y)-f(x) es\r\n-- menor que a. Por ejemplo,\r\n--    limite (\\n -> (2*n+1)\/(n+5)) 0.001  ==  1.9900110987791344\r\n--    limite (\\n -> (1+1\/n)**n) 0.001     ==  2.714072874546881\r\n-- ---------------------------------------------------------------------\r\n\r\nlimite :: (Num a, Enum a, Num b, Ord b) => (a -> b) -> b -> b\r\nlimite f a = \r\n    head [f x | x <- [1..],\r\n                maximum [abs(f y - f x) | y <- [x+1..x+100]] < a]\r\n\r\n-- ---------------------------------------------------------------------\r\n-- Ejercicio 3. Calcular el l\u00edmite de la sucesi\u00f3n de Ramanujan con una\r\n-- aproximaci\u00f3n de 10^(-20).\r\n-- ---------------------------------------------------------------------\r\n\r\n-- El c\u00e1lculo es\r\n--    ghci> limite ramanujan 1e-20\r\n--    3.0\r\n\r\n-- ---------------------------------------------------------------------\r\n-- Ejercicio 4. Definir la funci\u00f3n\r\n--    dibujoRamanujan :: Double -> IO ()\r\n-- tal que (dibujoRamanujan n) dibuja los n primeros t\u00e9rminos de la\r\n-- sucesi\u00f3n de Ramanujan. Por ejemplo, con (dibujoRamanujan 20) se\r\n-- obtiene la figura 1 (en el final de la entrada).\r\n-- ---------------------------------------------------------------------\r\n\r\ndibujoRamanujan :: Double -> IO ()\r\ndibujoRamanujan n = \r\n    plotList [] [(m,ramanujan m) | m <- [1..n]] \r\n\r\n-- Para dibujarla en el fichero Ramanujan.png\r\ndibujoRamanujan2 :: Double -> IO ()\r\ndibujoRamanujan2 n = \r\n    plotList [Title \"Sucesion de Ramanujan\",\r\n              PNG \"Ramanujan.png\",\r\n              YLabel \"Ramanujan(n)\", \r\n              Key Nothing] \r\n             [(m,ramanujan m) | m <- [1..n]] \r\n<\/pre>\n<figure id=\"attachment_3667\" aria-describedby=\"caption-attachment-3667\" style=\"width: 640px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/vestigium\/wp-content\/uploads\/2013\/09\/Ramanujan.png?ssl=1\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/vestigium\/wp-content\/uploads\/2013\/09\/Ramanujan.png?resize=640%2C480&#038;ssl=1\" alt=\"Figura 1\" width=\"640\" height=\"480\" class=\"size-full wp-image-3667\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/vestigium\/wp-content\/uploads\/2013\/09\/Ramanujan.png?w=640&amp;ssl=1 640w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/vestigium\/wp-content\/uploads\/2013\/09\/Ramanujan.png?resize=300%2C225&amp;ssl=1 300w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/vestigium\/wp-content\/uploads\/2013\/09\/Ramanujan.png?resize=150%2C112&amp;ssl=1 150w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/vestigium\/wp-content\/uploads\/2013\/09\/Ramanujan.png?resize=400%2C300&amp;ssl=1 400w\" sizes=\"(max-width: 640px) 100vw, 640px\" data-recalc-dims=\"1\" \/><\/a><figcaption id=\"caption-attachment-3667\" class=\"wp-caption-text\">Figura 1<\/figcaption><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>El viernes pasado, John D. Cook escribi\u00f3 el art\u00edculo Ramanujan\u2019s nested radical en el que comenta el siguiente problema planteado por Ramanujan en el Journal of Indian Mathematical Society: \u00bfCu\u00e1l es el valor de la siguiente expresi\u00f3n? A partir del problema he elaborado la siguiente relaci\u00f3n de ejercicios en Haskell para la asignatura de Inform\u00e1tica&#8230;<\/p>\n","protected":false},"author":2,"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":[5],"tags":[270,279],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_likes_enabled":false,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/3649"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/comments?post=3649"}],"version-history":[{"count":10,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/3649\/revisions"}],"predecessor-version":[{"id":3671,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/3649\/revisions\/3671"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=3649"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=3649"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=3649"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}