{"id":1777,"date":"2014-01-19T14:21:33","date_gmt":"2014-01-19T21:21:33","guid":{"rendered":"http:\/\/mattfife.com\/?p=1777"},"modified":"2014-01-19T14:22:24","modified_gmt":"2014-01-19T21:22:24","slug":"1777","status":"publish","type":"post","link":"https:\/\/mattfife.com\/?p=1777","title":{"rendered":"Summing all positive integers = -1\/12"},"content":{"rendered":"<p>What if you sat down and added up:<br \/>\n1+2+3+4+5&#8230;<br \/>\nall the way to infinity. What would you get? Most people would say &#8211; well &#8211; infinity.<br \/>\nWhat if I told you it wasn&#8217;t infinity &#8211; but -1\/12. Hogwash you&#8217;d say!<br \/>\nBut here he is proving it, and he claims the result is used all over in string theory and astronomical physics.<br \/>\n<object width=\"560\" height=\"315\" classid=\"clsid:d27cdb6e-ae6d-11cf-96b8-444553540000\" codebase=\"http:\/\/download.macromedia.com\/pub\/shockwave\/cabs\/flash\/swflash.cab#version=6,0,40,0\"><param name=\"allowFullScreen\" value=\"true\" \/><param name=\"allowscriptaccess\" value=\"always\" \/><param name=\"src\" value=\"\/\/www.youtube.com\/v\/w-I6XTVZXww?version=3&amp;hl=en_US&amp;rel=0\" \/><param name=\"allowfullscreen\" value=\"true\" \/><\/object><\/p>\n<p>How? Well, it has to do with how you think about and sum infinite series.<\/p>\n<p>Spoilers:<br \/>\nI remember a prof telling me that depending on how you interpret infinity &#8211; you can get systems in which parallel lines converge or diverge or stay parallel at infinity.\u00a0 It didn&#8217;t make a lot of sense at the time, but now you see a real case.<\/p>\n<p>You start to enter the rabbit hole when he proves the Grandi&#8217;s Series = 1\/2:<br \/>\n1-1+1-1+1&#8230;=1\/2.<\/p>\n<p>Here&#8217;s a link to all those explanations<br \/>\n<a href=\"http:\/\/periodicvideos.blogspot.co.uk\/2014\/01\/thanks.html\" target=\"_blank\" rel=\"nofollow nofollow\">http:\/\/periodicvideos.blogspot.co.uk\/2014\/01\/thanks.html<\/a><br \/>\nBut how does it really work?\u00a0 <a href=\"http:\/\/terrytao.wordpress.com\/2010\/04\/10\/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation\/\" target=\"_blank\">TerryTao<\/a> goes into tremendous detail of how the summation works and why the Grandi series can sum 1-1+1-1+1&#8230; even though it does NOT converge in the classic sense. But that&#8217;s the difference between applied mathematics\/physics and theoretical.\u00a0 In applied fields &#8211; you really DON&#8217;T have infinite things summing &#8211; you have real things &#8211; and it makes modeled results&#8230;unexpected.\u00a0 Amazingly enough &#8211; we had models for these systesms before we even knew of real things that displayed the properties.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What if you sat down and added up: 1+2+3+4+5&#8230; all the way to infinity. What would you get? Most people would say &#8211; well &#8211; infinity. What if I told you it wasn&#8217;t infinity &#8211; but -1\/12. Hogwash you&#8217;d say! But here he is proving it, and he claims the result is used all over in string theory and astronomical physics. How? Well, it has to do with how you think about and sum infinite series. Spoilers: I remember a&#8230;<\/p>\n<p class=\"read-more\"><a class=\"btn btn-default\" href=\"https:\/\/mattfife.com\/?p=1777\"> Read More<span class=\"screen-reader-text\">  Read More<\/span><\/a><\/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,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[9],"tags":[],"class_list":["post-1777","post","type-post","status-publish","format-standard","hentry","category-cool"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/s4WECr-1777","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/mattfife.com\/index.php?rest_route=\/wp\/v2\/posts\/1777","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mattfife.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mattfife.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mattfife.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mattfife.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1777"}],"version-history":[{"count":3,"href":"https:\/\/mattfife.com\/index.php?rest_route=\/wp\/v2\/posts\/1777\/revisions"}],"predecessor-version":[{"id":1780,"href":"https:\/\/mattfife.com\/index.php?rest_route=\/wp\/v2\/posts\/1777\/revisions\/1780"}],"wp:attachment":[{"href":"https:\/\/mattfife.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1777"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mattfife.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1777"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mattfife.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1777"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}