[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-573076-105":53,"doc-detail-573076-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","series-cheat-sheet-tests-theorem","SERIES CHEAT SHEET - Tests - Theorem","","SERIES CHEAT SHEET by Blake Farhman (Lafayette College) compiles core convergence and divergence criteria for infinite series. It summarizes key tests used to determine whether sum of terms an converges, including nth term/divergence, integral test, comparison and limit comparison tests, alternating series test, ratio test, and root test. The sheet also lists common convergent series (geometric and p-series) and common divergent series (harmonic, and geometric with weaker ratio conditions).",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/series-cheat-sheet-tests-theorem/573076/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/series-cheat-sheet-tests-theorem/573076.png","ImageObject",442,249,{"name":88,"@type":89},"Genevieve","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":76},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What does the nth term (divergence) test say about a series?","Question",{"text":108,"@type":109},"If the sequence an does not converge to 0, then the series sum an diverges.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"When is the integral test applicable, and what is the conclusion?",{"text":113,"@type":109},"If there exists an integer N and a positive, continuous, decreasing function f such that f(n)=an for all n≥N, then sum an converges if and only if the improper integral from N to infinity of f(x) dx converges.",{"name":115,"@type":106,"acceptedAnswer":116},"How do the ratio test and root test determine convergence?",{"text":117,"@type":109},"Both use a limit: the ratio test compares L to 1 and the root test compares L to 1. If L\u003C1, sum an converges absolutely; if L>1, sum an diverges; if L=1, the test is inconclusive.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},573076,1792403707,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":76,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":73,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":9},1374391974585,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","SERIES CHEAT SHEET  \nBLAKE FARMAN  \nLafayette College  \nTests  \nTheorem (nth Term Test for Divergence) . If the sequence fang does not converge to zero, then the series Pan diverges.  \nWARNING: If limn!1 an = 0 then the test is inconclusive!  \nTheorem (The Integral Test) . Let fang be a sequence. Assume that there exists an integer N 􀀕 0 anda function f such that for all x 􀀕 N and for all n 􀀕 N  \n[1.](1. an)[ a](1. an)[n](1. an) > 0,  \n2. f (n) = an , and  \n3. f is  \n􀀏 positive,  \n􀀏 continuous, and  \n􀀏 decreasing  \nThen the series P an converges if and only if R11 f (x) dx converges.  \nTheorem (The Comparison Tests) . Let fang and fbng be sequences, and assume there exists some number N such that  \n0 \u003C an 􀀔 bn  \nis satis􀀌ed whenever n 􀀕 N .  \n(i) If P an diverges, then P bn also diverges.  \n(ii) If P bn converges, then P an also converges.  \nTheorem (The Limit Comparison Test) . Let fang and fbng be sequences, and assume there exists some number N such that  \n0 \u003C an ; bn  \nis satis􀀌ed whenever n 􀀕 N . If there exists some number c > 0 such that  \nan lim ~~ ~~ = c > 0 n!1 bn  \nthen either  \n􀀏 Pan and P bn both converge, or  \n􀀏 P an and P bn both diverge.  \n2 SERIES CHEAT SHEET  \nTheorem (Alternating Series Test) . Let fbng be a sequence. If there exists some N such that for all n 􀀔 N  \n(1) 0 \u003C bn  \n(2) bn+1 􀀔 bn  \n(3) limn!1 bn = 0 then the Alternating Series  \n1  \nX (􀀀1)n􀀀1bn  \nn=1  \nconverges.  \nTheorem (Ratio Test) . Let fang be a sequence and let  \nL = nl 􀀌 an+1a~~ ~~n 􀀌 :  \n1  \n(i) If L \u003C 1, then X an converges absolutely.  \nn=1  \n1  \n(ii) If 1 \u003C L 􀀔 1, then X an diverges.  \nn=1  \n(iii) If L = 1, then the test is inconclusive.  \nTheorem (Root Test) . Let fang be a sequence and let  \nL = nl pn jan j  :  \n1  \n(i) If L \u003C 1, then X an converges absolutely.  \nn=1  \n1  \n(ii) If 1 \u003C L 􀀔 1, then X an diverges.  \nn=1  \n(iii) If L = 1, then the test is inconclusive.  \nKnown Convergent Series  \nGeometric Series, jrj \u003C 1: 1X arn􀀀1 = a + ar + ar2 + : : : + arn􀀀1 + : : : = 1~~ ~~~~ ~~r  \nn=1  \np-series, 1 \u003C p:  1np = 1 + 12p + 31p + : : : + 1np + : : :  \nKnown Divergent Series  \nHarmonic Series:  1n = 1 + 12 + 13 + : : : + 1n + : : :  \n1  \nGeometric Series, 1 􀀔 jrj: X arn􀀀1 = a + ar + ar2 + ar3 + : : : + arn􀀀1 + : : :  \nn=1  \np-series, p 􀀔 1: 1X 1np = 1 + 12p + 31p + : : : + 1np + : : :","cbCaieM64CEXRINz","https://ap.wps.com/l/cbCaieM64CEXRINz","pdf",131409,"English","# Tests\n## nth Term Test for Divergence\n## The Integral Test\n## The Comparison Tests\n## The Limit Comparison Test\n## Alternating Series Test\n## Ratio Test\n## Root Test\n# Known Series","[{\"question\":\"What does the nth term (divergence) test say about a series?\",\"answer\":\"If the sequence an does not converge to 0, then the series sum an diverges.\"},{\"question\":\"When is the integral test applicable, and what is the conclusion?\",\"answer\":\"If there exists an integer N and a positive, continuous, decreasing function f such that f(n)=an for all n≥N, then sum an converges if and only if the improper integral from N to infinity of f(x) dx converges.\"},{\"question\":\"How do the ratio test and root test determine convergence?\",\"answer\":\"Both use a limit: the ratio test compares L to 1 and the root test compares L to 1. If L\\u003c1, sum an converges absolutely; if L\\u003e1, sum an diverges; if L=1, the test is inconclusive.\"}]","SERIES CHEAT SHEET - Tests - Theorem | PDF",1791931864]