{"id":234620,"date":"2025-06-14T10:26:01","date_gmt":"2025-06-14T10:26:01","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=234620"},"modified":"2025-06-14T10:26:02","modified_gmt":"2025-06-14T10:26:02","slug":"if-an-and-bn-are-divergent-then-an-bn-is-divergent-b-if-an-and-bn-are-divergent-then-an-bn-is-divergent-c-if-an-is-convergent-and-bn-is-divergent-then-an-bn-is-divergent-d","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/14\/if-an-and-bn-are-divergent-then-an-bn-is-divergent-b-if-an-and-bn-are-divergent-then-an-bn-is-divergent-c-if-an-is-convergent-and-bn-is-divergent-then-an-bn-is-divergent-d\/","title":{"rendered":") If {an} and {bn} are divergent, then {an bn} is divergent. b) If {an} and {bn} are divergent, then {an, bn} is divergent. c) If {an} is convergent and {bn} is divergent, then {an bn} is divergent. d) If {an} is convergent, then {an} is convergent."},"content":{"rendered":"\n

) If {an} and {bn} are divergent, then {an bn} is divergent. b) If {an} and {bn} are divergent, then {an, bn} is divergent. c) If {an} is convergent and {bn} is divergent, then {an bn} is divergent. d) If {an} is convergent, then {an} is convergent.<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Correct Answer: c) If {an} is convergent and {bn} is divergent, then {an\u202fbn} is divergent.<\/strong><\/p>\n\n\n\n

\n\n\n\n

A sequence is a list of numbers in a specific order, and a sequence is called convergent<\/em> if its terms approach a fixed value as the index increases. A sequence is divergent<\/em> if it does not approach a finite value.<\/p>\n\n\n\n

Statement c<\/strong> asserts that if one sequence, {an}, is convergent and the other, {bn}, is divergent, then the product sequence {an\u202fbn} must be divergent. This statement is true<\/strong>, and this can be confirmed by the definition of convergence and divergence.<\/p>\n\n\n\n

To illustrate, suppose {an} converges to a finite number L<\/em>, and {bn} does not settle on any finite value. The product {an\u202fbn} will behave erratically if {bn} grows without bound or oscillates, and therefore, {an\u202fbn} will also fail to approach a single finite number. For example, if {an} converges to 1 and {bn} is the sequence {n}, then {an\u202fbn} = {n}, which diverges to infinity. If {bn} = (\u20131)^n, an oscillating divergent sequence, then even if {an} = 1, the product remains oscillating and hence divergent.<\/p>\n\n\n\n

Statement a<\/strong> is false because two divergent sequences can have a product that converges. Consider {an} = (\u20131)^n and {bn} = (\u20131)^n; both diverge, but their product {an\u202fbn} = 1 for all n, which converges.<\/p>\n\n\n\n

Statement b<\/strong> is vague, as {an, bn} suggests combining the sequences, perhaps as a tuple or interleaving them, which does not lead to a clear convergence analysis without more definition.<\/p>\n\n\n\n

Statement d<\/strong> is trivial and always true, but also circular in logic. Saying a convergent sequence is convergent does not add new information.<\/p>\n\n\n\n

Thus, option c<\/strong> is the best and most accurate logical statement.<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

) If {an} and {bn} are divergent, then {an bn} is divergent. b) If {an} and {bn} are divergent, then {an, bn} is divergent. c) If {an} is convergent and {bn} is divergent, then {an bn} is divergent. d) If {an} is convergent, then {an} is convergent. The Correct Answer and Explanation is: Correct Answer: […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-234620","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234620","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=234620"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234620\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=234620"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=234620"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=234620"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}