{"id":1427,"date":"2017-11-13T23:00:24","date_gmt":"2017-11-14T04:00:24","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1427"},"modified":"2017-11-13T23:00:24","modified_gmt":"2017-11-14T04:00:24","slug":"hw-8-challenge-3ab","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/11\/13\/hw-8-challenge-3ab\/","title":{"rendered":"HW 8: Challenge 3(a,b)"},"content":{"rendered":"

Consider the sequence of functions \"g_n:[0,\infty]\to defined by \"g_n(x):=\frac{x^n}{n}\"<\/p>\n

a) Prove that \"(g_n)\" converges uniformly on \"[0,1]\" and find \"g=\lim.<\/p>\n

\"(g_n)=(\frac{x^n}{n})=\{x,<\/p>\n

In order to better examine this, look at \"x=.5\", then\u00a0\"(g_n)=(\frac{.5^n}{n})=\{.5,. The limit of \"\frac{.5^n}{n}\" as \"n\to\infty\" is 0. This holds similarly for \"x\in(0,1)\"<\/p>\n

For \"x=0, and\u00a0\"x=1,, so \"(g_n)\" converges pointwise to \"g(x)=0=\lim<\/p>\n

Let \"\epsilon>0\" be arbitrary and choose \"\frac{1}{N}<\epsilon\" (which exists by Archimedean Property). Let \"n\geq both be arbitrary.<\/p>\n

Examine \"|g_n(x)-g(x)|=|g_n(x)-0|=|\frac{x^n}{n}|\". Notice that \"|\frac{x^n}{n}|\leq, since \"n\geq. Thus, since \"\epsilon, were arbitrary (within the stated constraints), for all \"\epsilon>0,\exists such that \"g_n(x)-g(x)|<\epsilon\" whenever \"n and \"x\in. Thus by definition of unform converges,\u00a0\"(g_n)\" converges uniformly on \"[0,1]\".<\/p>\n

b) Show that \"g=\lim is differentiable on \"[0,1]\" and compute \"g'(x)\".<\/p>\n

As shown in part a)\u00a0\"g=\lim.<\/p>\n

Let \"c\in[0,1]\" be arbitrary and consider \"\lim_{x\to. Since \"g\" is constant and defined for \"c\", this is the same as\u00a0\"\lim_{x\to. This however, if the definition of the derivative of g at c, which therefore exists. Since \"c was arbitrary, the derivative \"g'(x)=0\" exists for all points in \"[0,1]\", and thus by definition of differentiable,\"g'(x)=0\" is differentiable on\u00a0\"[0,1]\".<\/p>\n","protected":false},"excerpt":{"rendered":"

Consider the sequence of functions defined by a) Prove that converges uniformly on and find . In order to better examine this, look at , then\u00a0. The limit of as is 0. This holds similarly for For and\u00a0, so converges pointwise to Let be arbitrary and choose (which exists by Archimedean Property). Let both be […]<\/p>\n","protected":false},"author":3527,"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":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[58827,58826],"tags":[],"class_list":["post-1427","post","type-post","status-publish","format-standard","hentry","category-challenge-solutions","category-peer-grading"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-n1","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1427","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/users\/3527"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1427"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1427\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1427"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1427"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1427"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}