[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-573078-105":53,"doc-detail-573078-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-convergencedivergence-flow-chart-test-for-divergence","Series Convergence/Divergence Flow Chart - Test for Divergence","","Flow chart and testing guide for determining convergence or divergence of infinite series. Covers core methods including geometric, alternating, telescoping, and Taylor series, plus limit-based criteria and interval of convergence checks. Provides decision steps for comparing sequences (comparison test), integrating functions (integral test), and applying ratio and root tests to terms. Includes a problem set referencing Stewart’s Calculus (Problems 1–38) to practice the tests on varied series forms.",{"@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":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/series-convergencedivergence-flow-chart-test-for-divergence/573078/",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-convergencedivergence-flow-chart-test-for-divergence/573078.png","ImageObject",442,249,{"name":88,"@type":89},"Aurelia","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":79},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How does the flow chart decide divergence for a geometric series?","Question",{"text":108,"@type":109},"It uses the geometric series parameter r and checks whether |r| \u003C 1. If |r| \u003C 1 it converges; otherwise it diverges.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What tests are included besides geometric and alternating series?",{"text":113,"@type":109},"The guide includes telescoping series, Taylor series checks, comparison test, integral test, ratio test, and root test. It also includes limit-of-partial-sum reasoning for finiteness.",{"name":115,"@type":106,"acceptedAnswer":116},"Where can the practice problems be found?",{"text":117,"@type":109},"The document lists practice items as “Problems 1-38 from Stewart’s Calculus, page 784,” covering multiple series to apply the listed convergence/divergence tests.","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},573078,1792172375,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":79,"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},1099514068365,"https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068","Series Convergence/Divergence Flow Chart  \nTEST FOR DIVERGENCE  \nDoes limn→∞ an = 0? ~~ NO ~~  \n~~p~~-~~SERIES ~~~~ ~~  Pan Converges  \n Does an = 1/np , n ≥ 1?  YES    \n Pan Diverges  \nGEOMETRIC SERIES  Pan = 1r  \n  YES    \nDoes an = arn􀀀1 , n ≥ 1?  Is | r | \u003C 1?   \nPan Diverges  \nALTERNATING SERIES  \nPan Diverges  \nYES  \nIs p > 1?  \nYES  \n  NO    \nYES  \nNO  \nDaon a(1=)n1bn)n, bbnn  0? ~~ ~~YES~~  ~~ Is bn+1 ≤ bn & nl bn = 0?   \nNO  \nTELESCOPING SERIES  \n\n| Do subsequent terms cancel out previous terms in the sum? May have to use partial fractions, properties of logarithms, [etc. to](etc. to) put into appropriate form. | YES | Does lim sn = sn→∞ s 􀀌nite? |\n| --- | --- | --- |\n|  |  |  |\n\nTAYLOR SERIES  \n\n| Does an = f~~ ~~()!(a) (x − a)n? |  | \u003Cbr>Is x in interval of convergence? |\n| --- | --- | --- |\n|  |  |  |\n|  |  |  |\n\nYES  \n  YES    \n  NO    \n NO  \nTry one or more of the following tests:  \nCOMPARISON TEST  \n  YES ~~ ~~ Is 0 ≤ an ≤ bn? ~~  ~~YES~~    ~~ Pan Converges  \nNO    \n Is 0 ≤ bn ≤ an? ~~  ~~YES~~  ~~ Pan Diverges  \n| | Does | ∞\u003Cbr>X bn converge? n=1 |\n| --- | --- | --- |\n\nYES  \nINTEGRAL TEST  \nNO  \n Du[ooa,,)anp?o=sitifv(en)&, fd(xec)riseascontinginon- ~~ ~~YES~~    ~~ Does Za∞ f (x)dx converge?   \nRATIO TEST  \nIs nl 􀀌 ana 􀀌 \u003C 1?  \nROOT TEST  \nProblems 1-38 from Stewart’s Calculus, page 784  \n1. ∞ n2X n2~~ ~~1nn=1  \n2.  nn2~~ ~~1n  \n3.  n2~~ ~~~~ ~~n  \n4. (−1)n􀀀1nn2~~ ~~1n  \n5. ∞X (~~ ~~−323)nn+1 n=1  \n6.  􀀒 ~~ ~~1~~ ~~n8n 􀀓 n  \n7. ∞X  1  n=2 np ln(n)  \n8. k (k2k2!~~ ~~)!  \n∞  \n9. X k2 e 􀀀k  \nk=1  \n∞  \n10. X n2 e 􀀀n3 n=1  \n11.  (1ln))1  \n12. ∞X(−1)nn2~~ ~~~~ ~~25 n=1  \n∞ 3nn2  \n13. X  n!  n=1  \n∞  \n14. X sin(n) n=1  \n∞ n!  \n15.  2 · 5 · 8 · ··· · (3n + 2)  \n16.  n2n3~~ ~~~~ ~~11  \n∞  \n17. X (−1)n 21/nn=1  \n18.  (1)~~n~~􀀀11  \n19. ∞X(−1)n l(n~~n~~) n=1  \n20.  k~~ ~~+5k~~ ~~5  \n21.  (~~ ~~−2nn)2n  \n∞ √ n2 − 1 22. X n3 + 2n2 + 5  \nn=1  \n∞  \n23. X tan(1/n) n=1  \n24.  cons2/42n)  \n25. ∞ ~~ ~~nX en!2 n=1  \n∞ n2 + 1  \n26. X  5n n=1  \n27. ∞X (kklk1))3 k=1  \n28.  e1n/2n ∞ tan􀀀1(n)  \n29.   n √ n   \n30. (−1)j j~~ ~~5  \n31. k 3k~~ ~~4k  \n32.  (2nn2n)n ∞ sin(1/n)  \n33.   √ n   \n34.  n~~ ~~+~~ ~~n~~ ~~1cos2~~ ~~(n)  \n35.  􀀒 n~~ ~~~~ ~~1 􀀓 n2  \n36.  (ln(n1))ln(n)  \n∞  \n37. X ( 2 − 1)nn=1  \n∞  \n38. X ( 2 − 1) n=1","cbCaihNN4RaZLPLZ","https://ap.wps.com/l/cbCaihNN4RaZLPLZ","pdf",48647,"English","# Series Convergence/Divergence Tests\n## Core series types and criteria\n## Comparison, integral, ratio, and root tests\n## Practice problems (Stewart’s Calculus 1–38)","[{\"question\":\"How does the flow chart decide divergence for a geometric series?\",\"answer\":\"It uses the geometric series parameter r and checks whether |r| \\u003c 1. If |r| \\u003c 1 it converges; otherwise it diverges.\"},{\"question\":\"What tests are included besides geometric and alternating series?\",\"answer\":\"The guide includes telescoping series, Taylor series checks, comparison test, integral test, ratio test, and root test. It also includes limit-of-partial-sum reasoning for finiteness.\"},{\"question\":\"Where can the practice problems be found?\",\"answer\":\"The document lists practice items as “Problems 1-38 from Stewart’s Calculus, page 784,” covering multiple series to apply the listed convergence/divergence tests.\"}]","Series Convergence/Divergence Flow Chart - Test for Divergence | PDF",1791931865]