{"id":234624,"date":"2025-06-14T10:28:58","date_gmt":"2025-06-14T10:28:58","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=234624"},"modified":"2025-06-14T10:29:02","modified_gmt":"2025-06-14T10:29:02","slug":"let-an-and-bn-be-series-with-positive-terms","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/14\/let-an-and-bn-be-series-with-positive-terms\/","title":{"rendered":"Let #an and #bn be series with positive terms."},"content":{"rendered":"\n

Let #an and #bn be series with positive terms. Is each of the following statements true or false? If the statement is false, give an example that disproves the statement. (a) If #an and #bn are divergent, then #an bn is divergent. (b) If #an converges and #bn diverges, then #an bn diverges. (c) If #an and #bn are convergent, then #an bn is convergent<\/p>\n\n\n\n

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

ChatGPT said:<\/h6>\n\n\n\n

Correct Answers:<\/strong><\/p>\n\n\n\n

(a) False<\/strong>
(b) False<\/strong>
(c) True<\/strong><\/p>\n\n\n\n

\n\n\n\n

Explanation (300 words, no use of the word we<\/em>):<\/h3>\n\n\n\n

Consider each statement involving infinite series with positive terms<\/strong>:<\/p>\n\n\n\n

(a) If \u2211an and \u2211bn are divergent, then \u2211an\u00b7bn is divergent.<\/strong>
This is false<\/strong>. A counterexample proves the claim incorrect. Let an = 1\/n and bn = 1\/n. Both \u2211(1\/n) diverge (harmonic series). However, the product series an\u00b7bn = (1\/n)\u00b7(1\/n) = 1\/n\u00b2. The series \u2211(1\/n\u00b2) converges<\/strong>. This contradicts the statement. Therefore, divergence of two series does not guarantee divergence of their term-by-term product.<\/p>\n\n\n\n

(b) If \u2211an converges and \u2211bn diverges, then \u2211an\u00b7bn diverges.<\/strong>
This is also false<\/strong>. An example disproves the statement. Let an = 1\/n\u00b2 (convergent p-series with p = 2) and bn = 1 (constant sequence). Then an\u00b7bn = 1\/n\u00b2, and \u2211(1\/n\u00b2) converges. Here, \u2211an\u00b7bn converges even though \u2211bn diverges (since \u22111 diverges). Therefore, convergence of one series and divergence of the other does not necessarily imply divergence of their product.<\/p>\n\n\n\n

(c) If \u2211an and \u2211bn are convergent, then \u2211an\u00b7bn is convergent.<\/strong>
This is true<\/strong> for series with positive terms<\/strong>. When an \u2265 0 and bn \u2265 0, and both \u2211an and \u2211bn converge, the term-by-term product an\u00b7bn tends toward 0. Since an and bn are bounded and positive, the product an\u00b7bn is also bounded and positive. The comparison test can be used to compare the product series to another known convergent series. For example, since an is convergent, it is bounded by some M > 0. Then an\u00b7bn \u2264 M\u00b7bn, and since \u2211bn converges, the comparison test confirms convergence of \u2211an\u00b7bn.<\/p>\n\n\n\n

In conclusion: Statement (a) is false, proven by the harmonic series; statement (b) is false, shown using convergent p-series; statement (c) is true under the condition of positive terms.<\/p>\n\n\n\n

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

Let #an and #bn be series with positive terms. Is each of the following statements true or false? If the statement is false, give an example that disproves the statement. (a) If #an and #bn are divergent, then #an bn is divergent. (b) If #an converges and #bn diverges, then #an bn diverges. (c) If […]<\/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-234624","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234624","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=234624"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234624\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=234624"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=234624"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=234624"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}