[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-573075-105":53,"doc-detail-573075-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","harolds-series-convergence-tests-cheat-sheet-22-september-2025","Harold’s Series Convergence Tests Cheat Sheet - 22 September 2025","","Harold’s Series Convergence Tests Cheat Sheet provides a compact reference for determining whether infinite series converge or diverge. It summarizes multiple criteria including Divergence/Nth term, Geometric, p-Series, Alternating series (with remainder bound), Integral, Ratio, Root, Direct comparison, Limit comparison, and Telescoping series. For each test, it states the relevant series form and the exact conditions that guarantee convergence or divergence, plus brief notes on when a test is inconclusive or needs algebraic rewriting.",{"@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/harolds-series-convergence-tests-cheat-sheet-22-september-2025/573075/",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/harolds-series-convergence-tests-cheat-sheet-22-september-2025/573075.png","ImageObject",442,249,{"name":88,"@type":89},"River Wang","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":47},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"Which convergence test cannot be used to show convergence?","Question",{"text":108,"@type":109},"The Divergence or nth Term Test is stated as not usable to show convergence; it only applies to divergence when the nth term fails the necessary condition.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What is the convergence condition for a geometric series?",{"text":113,"@type":109},"For a geometric series, convergence requires the absolute value of the common ratio to be less than 1, i.e., |r| \u003C 1, along with the corresponding sum formula.",{"name":115,"@type":106,"acceptedAnswer":116},"How do the Ratio and Root tests determine inconclusive cases?",{"text":117,"@type":109},"The Ratio test is inconclusive if the limit of |a_{n+1}/a_n| equals 1. The Root test is inconclusive when the limit of the n-th root of |a_n| equals 1.","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},573075,1792090388,{"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":47,"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},1099514067438,"https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542","Harold’s Series Convergence Tests Cheat Sheet  \n22 September 2025  \n\n| 1\u003Cbr>\u003Cbr>Divergence or nth Term Test\u003Cbr>\u003Cbr>Series: ∑=1 􀜽􀯡\u003Cbr>\u003Cbr>Condition(s) of Convergence: None. This test cannot be used to show convergence.\u003Cbr>Condition(s) of Divergence:\u003Cbr>l􀯡 􀜽􀯡 ≠ 0\u003Cbr>| 2\u003Cbr>Geometric Series Test\u003Cbr>Series: ∑=0 􀜽􀝎􀯡\u003Cbr>Condition of Convergence:|􀝎| \u003C 1\u003Cbr>Sum: 􀜁 = lim 􀯔 (1−􀯥􀳙 ) =  􀯔 􀯡→∞ 1−􀯥 1−􀯥\u003Cbr>Condition of Divergence:|􀝎| ≥ 1 | 3\u003Cbr>\u003Cbr>p-Series Test\u003Cbr>\u003Cbr>Series: ∑=1 􀯡1􀳛\u003Cbr>\u003Cbr>Condition of Convergence:\u003Cbr>\u003Cbr>\u003Cbr>Condition of Divergence:\u003Cbr>􀝌 ≤ 1\u003Cbr>\u003Cbr>􀝌 > 1 |\n| --- | --- | --- |\n| 4\u003Cbr>Alternating Series Test\u003Cbr>Series: ∑=1 (−1)􀯡+1 􀜽􀯡\u003Cbr>Condition of Convergence:\u003Cbr>0 \u003C 􀜽􀯡+1 ≤ 􀜽􀯡 l􀯡 􀜽􀯡 = 0\u003Cbr>or if ∑=0 |􀜽􀯡 | is convergent Condition of Divergence:\u003Cbr>None. This test cannot be used to show divergence.\u003Cbr>* Remainder: |􀜴􀯡 | ≤ 􀜽􀯡+1 | 5\u003Cbr> Integral Test \u003Cbr>\u003Cbr>Series: ∑=1 􀜽􀯡   when 􀜽􀯡 = 􀝂 (􀝊) ≥ 0 \u003Cbr> and 􀝂(􀝊) is continuous, positive and   decreasing \u003Cbr>\u003Cbr>Condition of Convergence:   ∫1∞ 􀝂 (􀝔)􀝀􀝔 converges \u003Cbr>\u003Cbr>Condition of Divergence:∫1∞ 􀝂 (􀝔)􀝀􀝔 diverges\u003Cbr>* Remainder: 0 \u003C 􀜴􀯇 ≤ ∫􀝂 (􀝔)􀝀􀝔 \u003Cbr>| 6\u003Cbr>Ratio Test\u003Cbr>Series: ∑=1 􀜽􀯡\u003Cbr>Condition of Convergence: lim 􀜽􀯡+1  | \u003C 1\u003Cbr>􀯡→∞ | 􀜽􀯡\u003Cbr>Condition of Divergence: lim |􀜽􀯡+1  | > 1\u003Cbr>􀯡→∞ 􀜽􀯡\u003Cbr>* Test inconclusive if lim |􀜽􀯡+1  | = 1\u003Cbr>􀯡→∞ 􀜽􀯡 |\n| 7\u003Cbr>\u003Cbr>Root Test\u003Cbr>Series: ∑=1 􀜽􀯡\u003Cbr>\u003Cbr>Condition of Convergence:\u003Cbr>\u003Cbr>lim 􀳙√|􀜽􀯡 | \u003C 1 􀯡→∞\u003Cbr>\u003Cbr>lim 􀳙√|􀜽􀯡 | > 1 􀯡→∞\u003Cbr>\u003Cbr>lim 􀳙√|􀜽􀯡 | = 1 􀯡→∞\u003Cbr>\u003Cbr>Condition of Divergence:\u003Cbr>\u003Cbr>* Test inconclusive if | 8\u003Cbr>Direct Comparison Test\u003Cbr>(􀜽􀯡 , 􀜾􀯡 > 0)\u003Cbr>Series: ∑=1 􀜽􀯡\u003Cbr>Condition of Convergence:\u003Cbr>0 \u003C 􀜽􀯡 ≤ 􀜾􀯡 and ∑=0 􀜾􀯡 is absolutely\u003Cbr>convergent\u003Cbr>Condition of Divergence:\u003Cbr>0 \u003C 􀜾􀯡 ≤ 􀜽􀯡 and ∑=0 􀜾􀯡 diverges | 9\u003Cbr>\u003Cbr>Limit Comparison Test\u003Cbr>({􀜽􀯡 }, {􀜾􀯡 } > 0)\u003Cbr>\u003Cbr>Series: ∑=1 􀜽􀯡\u003Cbr>\u003Cbr>Condition of Convergence:\u003Cbr>\u003Cbr>\u003Cbr>lim\u003Cbr>􀯡→∞\u003Cbr>􀜽􀯡 ~~ ~~ = 􀜮 > 0 􀜾􀯡\u003Cbr>\u003Cbr>and ∑=0 􀜾􀯡 converges\u003Cbr>\u003Cbr>Condition of Divergence:\u003Cbr>\u003Cbr>lim\u003Cbr>􀯡→∞\u003Cbr>􀜽􀯡 ~~ ~~ = 􀜮 > 0 􀜾􀯡\u003Cbr>\u003Cbr>and ∑=0 􀜾􀯡 diverges |\n| 10\u003Cbr>Telescoping Series Test\u003Cbr>Series: ∑=1 (􀜽􀝊+1 − 􀜽􀝊)\u003Cbr>Condition of Convergence: lim 􀜽􀝊 = 􀜮􀯡→∞\u003Cbr>Condition of Divergence: None | NOTE:\u003Cbr>1) May need to reformat with partial fraction expansion or log rules.\u003Cbr>2) Expand first 5 terms. n=1,2,3,4,5 .\u003Cbr>3) Cancel duplicates.\u003Cbr>4) Determine limit L by taking the limit as 􀝊 → ∞ .\u003Cbr>5) Sum: 􀜵 = 􀜽1 − 􀜮 | NOTE: These tests prove convergence and divergence, not the actual limit 􀜮 or sum S.\u003Cbr>Sequence: l􀯡 􀜽􀯡 = 􀜮\u003Cbr>(􀜽􀯡 , 􀜽􀯡+1, 􀜽􀯡+2,…)\u003Cbr>Series: ∑= 1 􀜽􀯡 = 􀜁\u003Cbr>(􀜽􀯡 + 􀜽􀯡+1 + 􀜽􀯡+2 + ⋯ ) |\n\nChoosing a Convergence Test for Infinite Series  \nCourtesy David J. Manuel","cbCaicvLc30UTDvz","https://ap.wps.com/l/cbCaicvLc30UTDvz","pdf",293918,"English","# Choosing a Convergence Test for Infinite Series\n## Divergence or nth Term Test\n## Geometric Series Test\n## p-Series Test\n## Alternating Series Test\n## Integral Test\n## Ratio Test\n## Root Test\n## Direct Comparison Test\n## Limit Comparison Test\n## Telescoping Series Test","[{\"question\":\"Which convergence test cannot be used to show convergence?\",\"answer\":\"The Divergence or nth Term Test is stated as not usable to show convergence; it only applies to divergence when the nth term fails the necessary condition.\"},{\"question\":\"What is the convergence condition for a geometric series?\",\"answer\":\"For a geometric series, convergence requires the absolute value of the common ratio to be less than 1, i.e., |r| \\u003c 1, along with the corresponding sum formula.\"},{\"question\":\"How do the Ratio and Root tests determine inconclusive cases?\",\"answer\":\"The Ratio test is inconclusive if the limit of |a_{n+1}/a_n| equals 1. The Root test is inconclusive when the limit of the n-th root of |a_n| equals 1.\"}]","Harold’s Series Convergence Tests Cheat Sheet - 22 September 2025 | PDF",1791931863]