{"id":32235,"date":"2021-04-22T09:02:55","date_gmt":"2021-04-22T08:02:55","guid":{"rendered":"http:\/\/kmr.dialectica.se\/wp\/?page_id=32235"},"modified":"2021-04-22T09:02:55","modified_gmt":"2021-04-22T08:02:55","slug":"convergence","status":"publish","type":"page","link":"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-one-real-variable\/basic-properties-of-functions\/convergence\/","title":{"rendered":"Convergence"},"content":{"rendered":"<p>his page is a sub-page of our page on <a href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-one-real-variable\/basic-properties-of-functions\/\" title=\"at the KMR web site\" target=\"_blank\" rel=\"noopener\">Basic properties of functions<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/ <\/p>\n<p><strong>The sub-pages of this page are<\/strong>: <\/p>\n<p>\u2022 &#8230; <\/p>\n<p>\/\/\/\/\/\/\/ <\/p>\n<p><strong>Related KMR-pages<\/strong>: <\/p>\n<p>\u2022 &#8230; <\/p>\n<p>\/\/\/\/\/\/\/ <\/p>\n<p><strong>Other relevant sources of information<\/strong>: <\/p>\n<p>\u2022 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Norm_(mathematics)\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Norm<\/a><br \/>\n\u2022 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Topology\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Topology<\/a><br \/>\n\u2022 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Metric_(mathematics)\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Metric<\/a><br \/>\n\u2022 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Pointwise_convergence\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Pointwise convergence<\/a><br \/>\n\u2022 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Uniform_convergence\" title=\"The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept, which was first formalized by Karl Weierstrass, is important because several properties of the functions f_n, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit f if the convergence is uniform, but not necessarily if the convergence is not uniform (Wikipedia)\" target=\"_blank\" rel=\"noopener\">Uniform convergence<\/a><br \/>\n\u2022 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Pointwise_convergence#Almost_everywhere_convergence\" title=\"at Wikipedia\" target=\"_blank\" rel=\"noopener\">Almost everywhere convergence<\/a><\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p>The interactive simulations on this page can be navigated with the <a href=\"http:\/\/www.pacifict.com\/FreeStuff.html\">Free Viewer<\/a><br \/>\nof the <a href=\"http:\/\/www.pacifict.com\/Home.html\">Graphing Calculator<\/a>. <\/p>\n<p>\/\/\/\/\/\/\/  <\/p>\n<p><strong>Uniform convergence<\/strong>:<\/p>\n<p><object classid=\"clsid:d27cdb6e-ae6d-11cf-96b8-444553540000\" width=\"640\" height=\"385\" 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=\"http:\/\/www.youtube.com\/v\/5j_CWup3iSg?fs=1&amp;hl=en_US&amp;rel=0\" \/><param name=\"allowfullscreen\" value=\"true\" \/><embed type=\"application\/x-shockwave-flash\" width=\"640\" height=\"385\" src=\"http:\/\/www.youtube.com\/v\/5j_CWup3iSg?fs=1&amp;hl=en_US&amp;rel=0\" allowscriptaccess=\"always\" allowfullscreen=\"true\"><\/embed><\/object> <\/p>\n<p>\/\/\/ <\/p>\n<p><iframe loading=\"lazy\" width=\"476\" height=\"455\" src=\"https:\/\/www.youtube.com\/embed\/c85xWA6SgEY\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><br \/>\n<a href=\"https:\/\/tinyurl.com\/h8tc92kb?download\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\">The interactive simulation that created this movie<\/a>.<\/p>\n<p><strong>Pointwise &#8211; but not uniform &#8211; convergence<\/strong>:<\/p>\n<p><iframe loading=\"lazy\" width=\"791\" height=\"455\" src=\"https:\/\/www.youtube.com\/embed\/gluTM7u83jE\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><br \/>\n<a href=\"https:\/\/tinyurl.com\/cpvwre3f\" title=\"in Ambj\u00f6rn's Confolio archive\" target=\"_blank\" rel=\"noopener\"> The interactive simulation that created this movie<\/a>. <\/p>\n<p>When <span class=\"wp-katex-eq\" data-display=\"false\"> \\, n \\rightarrow \\infty \\, <\/span> this sequence converges pointwise for each value of <span class=\"wp-katex-eq\" data-display=\"false\"> \\, x <\/span>, i.e., <span class=\"wp-katex-eq\" data-display=\"false\"> \\, f_n(x) \\rightarrow F(x) \\, <\/span> for each <span class=\"wp-katex-eq\" data-display=\"false\"> \\, x <\/span>. However, the convergence is not uniform, since there is no \u201dtail-value\u201d <span class=\"wp-katex-eq\" data-display=\"false\"> \\, N = N(\\epsilon) \\, <\/span> such that the \u201dtail\u201d of the sequence  <span class=\"wp-katex-eq\" data-display=\"false\"> \\, f_n(x) \\, <\/span> stays within <span class=\"wp-katex-eq\" data-display=\"false\"> \\, \\epsilon \\, <\/span> of the  pointwise limit function <span class=\"wp-katex-eq\" data-display=\"false\"> \\, F(x) \\, <\/span> for EVERY value of <span class=\"wp-katex-eq\" data-display=\"false\"> \\, n <\/span> that is greater than <span class=\"wp-katex-eq\" data-display=\"false\"> \\, N <\/span>. In order for the convergence to be uniform, the tail of the sequence <span class=\"wp-katex-eq\" data-display=\"false\"> \\, f_n(x) \\, <\/span> must stay within the epsilon-band of the limit function <span class=\"wp-katex-eq\" data-display=\"false\"> \\, F(x) \\, <\/span> FOR EVERY VALUE of <span class=\"wp-katex-eq\" data-display=\"false\"> \\, \\epsilon &gt; 0 <\/span>.<\/p>\n<p><strong>The disappearing wave<\/strong>: <\/p>\n<p>Let  <span class=\"wp-katex-eq\" data-display=\"false\"> \\, g_n(x) = \\dfrac{nx}{e^{nx}} \\, <\/span> be given by the red curve and consider the sequence <span class=\"wp-katex-eq\" data-display=\"false\"> \\, { \\{ g_n \\} }_{n=1}^{\\infty} <\/span>.<\/p>\n<p>Each function <span class=\"wp-katex-eq\" data-display=\"false\"> \\, g_n \\, <\/span> is continuous at the point <span class=\"wp-katex-eq\" data-display=\"false\"> \\, x = 0 <\/span>,<br \/>\nsince when <span class=\"wp-katex-eq\" data-display=\"false\"> \\, x \\rightarrow 0 \\, <\/span> we have <span class=\"wp-katex-eq\" data-display=\"false\"> \\, \\lim\\limits_{x \\to 0} g_n(x) = 0 = g_n(0) \\, <\/span> for each <span class=\"wp-katex-eq\" data-display=\"false\"> \\, n \\, <\/span>.<br \/>\nWhen <span class=\"wp-katex-eq\" data-display=\"false\"> \\, x = 1\/n \\, <\/span> we have <span class=\"wp-katex-eq\" data-display=\"false\"> \\, g_n(1\/n) = 1\/e <\/span>, which is the maximum value of <span class=\"wp-katex-eq\" data-display=\"false\"> \\, g_n <\/span>.<\/p>\n<p>The sequence of functions <span class=\"wp-katex-eq\" data-display=\"false\"> \\, { \\{ g_n \\} }_{n=1}^{\\infty} <\/span> behaves like a wave that \u201dcompresses itself\u201d towards the point <span class=\"wp-katex-eq\" data-display=\"false\"> \\, x = 0 \\, <\/span> and threatens to break at this point. Each member function <span class=\"wp-katex-eq\" data-display=\"false\"> \\,  g_n  \\, <\/span> attains its maximum amplitude of <span class=\"wp-katex-eq\" data-display=\"false\"> \\, 1\/e \\, <\/span> at the point <span class=\"wp-katex-eq\" data-display=\"false\"> \\, x = 1\/n <\/span>. The &#8220;sequence-wave&#8221; passes by each point <span class=\"wp-katex-eq\" data-display=\"false\"> \\, x &gt; 0 \\, <\/span> and then \u201ddies down\u201d towards amplitude <span class=\"wp-katex-eq\" data-display=\"false\"> \\, 0 \\, <\/span> at this point. Yet the wave never reaches the point <span class=\"wp-katex-eq\" data-display=\"false\"> \\, x = 0 <\/span>, because at this point it always has the amplitude <span class=\"wp-katex-eq\" data-display=\"false\"> \\, 0 \\, <\/span> since <span class=\"wp-katex-eq\" data-display=\"false\"> \\, g_n(0) = 0 \\, <\/span> for each value of <span class=\"wp-katex-eq\" data-display=\"false\"> \\, n  <\/span>.<\/p>\n<p>\/\/\/\/\/\/<\/p>\n","protected":false},"excerpt":{"rendered":"<p>his page is a sub-page of our page on Basic properties of functions. \/\/\/\/\/\/\/ The sub-pages of this page are: \u2022 &#8230; \/\/\/\/\/\/\/ Related KMR-pages: \u2022 &#8230; \/\/\/\/\/\/\/ Other relevant sources of information: \u2022 Norm \u2022 Topology \u2022 Metric \u2022 Pointwise convergence \u2022 Uniform convergence \u2022 Almost everywhere convergence \/\/\/\/\/\/\/ The interactive simulations on this &hellip; <a href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-one-real-variable\/basic-properties-of-functions\/convergence\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Convergence<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":2581,"menu_order":1,"comment_status":"open","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":["post-32235","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/32235","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/comments?post=32235"}],"version-history":[{"count":0,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/32235\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/2581"}],"wp:attachment":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/media?parent=32235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}