{"id":4379,"date":"2018-12-07T06:00:10","date_gmt":"2018-12-07T04:00:10","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=4379"},"modified":"2019-01-17T15:45:15","modified_gmt":"2019-01-17T13:45:15","slug":"aproximacion-entre-pi-y-e","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/aproximacion-entre-pi-y-e\/","title":{"rendered":"Aproximaci\u00f3n entre pi y e"},"content":{"rendered":"<p>El d\u00eda 11 de noviembre, se public\u00f3 en la cuenta de Twitter de <a href=\"http:\/\/bit.ly\/2DNfhzS\">Fermat&#8217;s Library<\/a> la siguiente curiosa identidad que relaciona los n\u00fameros e y pi:<br \/>\n<a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2018\/12\/Aproximacion_entre_pi_y_e.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2018\/12\/Aproximacion_entre_pi_y_e.png?resize=687%2C95\" alt=\"\" width=\"687\" height=\"95\" class=\"aligncenter size-full wp-image-4380\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2018\/12\/Aproximacion_entre_pi_y_e.png?w=687&amp;ssl=1 687w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2018\/12\/Aproximacion_entre_pi_y_e.png?resize=300%2C41&amp;ssl=1 300w\" sizes=\"(max-width: 687px) 100vw, 687px\" data-recalc-dims=\"1\" \/><\/a><\/p>\n<p>Definir las siguientes funciones:<\/p>\n<pre lang=\"text\">\n   sumaTerminos :: Int -> Double\n   aproximacion :: Double -> Int\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(sumaTerminos n) es la suma de los primeros n t\u00e9rminos de la serie 1\/(\u03c0\u00b2+ 1) + 1\/(4\u03c0\u00b2+1) + 1\/(9\u03c0\u00b2+1) + 1\/(16\u03c0\u00b2+ ) + &#8230; Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     sumaTerminos 10     ==  0.14687821811081034\n     sumaTerminos 100    ==  0.15550948345688423\n     sumaTerminos 1000   ==  0.15641637221314514\n     sumaTerminos 10000  ==  0.15650751113789382\n<\/pre>\n<ul>\n<li>(aproximaci\u00f3n x) es el menor n\u00famero de t\u00e9rminos que hay que sumar de la serie anterior para que se diferencie (en valor absoluto) de 1\/(e\u00b2-1) menos que x. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     aproximacion 0.1     ==  1\n     aproximacion 0.01    ==  10\n     aproximacion 0.001   ==  101\n     aproximacion 0.0001  ==  1013\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\n-- 1\u00aa definici\u00f3n de sumaTerminos\nsumaTerminos :: Int -> Double\nsumaTerminos n =\n  sum [1 \/ (((x ^ 2) * (pi ^ 2)) + 1) | x <- [1 .. fromIntegral n]]\n\n-- 2\u00aa definici\u00f3n de sumaTerminos\nsumaTerminos2 :: Int -> Double\nsumaTerminos2 0 = 0\nsumaTerminos2 n = 1 \/ (m^2 * pi^2 + 1) + sumaTerminos2 (n-1)\n  where m = fromIntegral n\n\n-- Definici\u00f3n de aproximacion\naproximacion :: Double -> Int\naproximacion x =\n  head [n | n <- [0..]\n          , abs (sumaTerminos n - 1 \/ (e^2 - 1)) < x]\n  where e = exp 1\n<\/pre>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\n\"S\u00f3lo s\u00e9 que no se nada\" conten\u00eda la jactancia de un excesivo saber, puesto que olvid\u00f3 a\u00f1adir: y aun de esto mismo no estoy completamente seguro.<\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>El d\u00eda 11 de noviembre, se public\u00f3 en la cuenta de Twitter de Fermat&#8217;s Library la siguiente curiosa identidad que relaciona los n\u00fameros e y pi: Definir las siguientes funciones: sumaTerminos :: Int -> Double aproximacion :: Double -> Int tales que (sumaTerminos n) es la suma de los primeros n t\u00e9rminos de la serie&#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":[5],"tags":[130,8,386,183,71,366,6],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4379"}],"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=4379"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4379\/revisions"}],"predecessor-version":[{"id":4580,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4379\/revisions\/4580"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=4379"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=4379"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=4379"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}