The integral test basically works from the definition of the integral (quick version: the integral is the accumulated sum of infinitely thin differential intervals ##dn## over a specified interval ##a->b##).<\/p>\n
A paraphrased version of the integral test<\/b> is as follows:<\/p>\n
Let there be a function ##f(n) = a_n## where ##a_n## is a series lying within the domain ##[k,oo)##. There exists another function ##f(x)## that is continuous<\/b>, positive<\/b>, and decreasing<\/b> such that the convergence or divergence of ##int_k^(oo)f(x)dx## determines the convergence or divergence of ##sum_(n=k)^(oo)a_n##, respectively.<\/p>\n
So, essentially, we have to integrate this, which is indeed continuous<\/b>, positive<\/b>, and decreasing<\/b> at ##[1,oo)##:<\/p>\n
##int_1^(oo)1\/(sqrt(x+1))dx##<\/p>\n
We can do that like so:<\/p>\n
##= int_1^(oo)(x+1)^(-“1\/2”)dx##<\/p>\n
##= |[2(x+1)^(“1\/2”)]|_1^(oo)##<\/p>\n
At this point we know that ##sqrt(x+1)## is a constantly increasing function, so it has an “open” accumulation that can never stop without a well-defined right-end boundary. Basically, it’s a half-open integral that extends its domain forever and so it has no finite area.<\/p>\n
##= 2(oo)^(“1\/2”) – cancel(2(1+1)^(“1\/2”))^”small”##<\/p>\n
##=> oo##<\/p>\n
The integral does not<\/b> converge, and so the series does not converge either. QED<\/p>\n","protected":false},"excerpt":{"rendered":"
The integral test basically works from the definition of the integral (quick version: the integral is the accumulated sum of infinitely thin differential intervals ##dn## over a specified interval ##a->b##). A paraphrased version of the integral test is as follows: Read More …<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"class_list":["post-67905","post","type-post","status-publish","format-standard","hentry"],"_links":{"self":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts\/67905","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/comments?post=67905"}],"version-history":[{"count":0,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts\/67905\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/media?parent=67905"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/categories?post=67905"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/tags?post=67905"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}