[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-572143-105":3,"detail-sidebar-cat-1-en-105":80,"doc-detail-572143-en":126},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","cheat-sheet-math-series-convergence-tests-1-10","Cheat Sheet - Math Series Convergence Tests - 1-10","","Cheat sheet summarizing core criteria for determining whether infinite series converge or diverge. Covers the nth-term divergence test, geometric series, p-series, alternating series, integral test, ratio test, root test, direct comparison and limit comparison tests, and telescoping series. Includes key conditions for convergence/divergence, common remainder estimates for alternating and integral tests, and practical calculation notes such as expanding initial terms and simplifying to obtain the limit behavior.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/presentations/","Presentations",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/cheat-sheet-math-series-convergence-tests-1-10/572143/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/cheat-sheet-math-series-convergence-tests-1-10/572143.png","ImageObject",442,249,{"name":42,"@type":43},"Malam Tenang","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":52,"interactionType":53,"userInteractionCount":30},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What does the nth-term test determine for a series?","Question",{"text":62,"@type":63},"If the nth term does not go to zero (an ≠ 0), the series diverges. If it does go to zero, the test is inconclusive and cannot confirm convergence.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"When can the geometric series be shown to converge?",{"text":67,"@type":63},"A geometric series converges when the absolute value of the common ratio is less than 1, i.e., |r| \u003C 1.",{"name":69,"@type":60,"acceptedAnswer":70},"How does the ratio test decide convergence or divergence?",{"text":71,"@type":63},"Compute lim n→∞ |a(n+1)|/|a(n)|. If the limit is less than 1, the series converges; if greater than 1, it diverges; if equal to 1, the test is inconclusive.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},572143,1792403968,{"code":4,"msg":81,"data":82},"success",[83,87,92,97,102,107,112,117,122],{"id":84,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":85,"slug":86},11,90,"presentations",{"id":88,"doc_module":22,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},12,"Resumes",80,"resumes",{"id":93,"doc_module":22,"doc_module_name":25,"category_name":94,"show_sort_weight":95,"slug":96},14,"Invoices",70,"invoices",{"id":98,"doc_module":22,"doc_module_name":25,"category_name":99,"show_sort_weight":100,"slug":101},15,"Posters",60,"posters",{"id":103,"doc_module":22,"doc_module_name":25,"category_name":104,"show_sort_weight":105,"slug":106},16,"Social Media",50,"social-media",{"id":108,"doc_module":22,"doc_module_name":25,"category_name":109,"show_sort_weight":110,"slug":111},17,"Forms",40,"forms",{"id":113,"doc_module":22,"doc_module_name":25,"category_name":114,"show_sort_weight":115,"slug":116},18,"Letters",30,"letters",{"id":118,"doc_module":22,"doc_module_name":25,"category_name":119,"show_sort_weight":120,"slug":121},21,"Paper Templates",5,"papers-templates",{"id":123,"doc_module":22,"doc_module_name":25,"category_name":124,"show_sort_weight":4,"slug":125},158,"General","general-158",{"code":4,"msg":81,"data":127},{"doc_id":78,"user_id":128,"nickname":42,"user_avatar":129,"doc_module":22,"category_id":84,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":30,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":26,"language":135,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":12,"update_tm":139,"read_time":22},962093532863,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","cheat sheet  \n\n| 1\u003Cbr>Divergence or nth Term Test\u003Cbr>Series: ∑=1 􀜽􀯡\u003Cbr>Condition(s) of Convergence: None. This test cannot be used to show convergence.\u003Cbr>Condition(s) of Divergence:\u003Cbr>l􀯡 􀜽􀯡 ≠ 0 | 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>p-Series Test\u003Cbr>Series: ∑=1 􀯡1􀳛\u003Cbr>Condition of Convergence:\u003Cbr>􀝌 > 1\u003Cbr>Condition of Divergence:\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>Series: ∑=1 􀜽􀯡\u003Cbr>when 􀜽􀯡 = 􀝂 (􀝊) ≥ 0\u003Cbr>and 􀝂(􀝊) is continuous, positive and decreasing\u003Cbr>Condition of Convergence:∫1∞ 􀝂 (􀝔)􀝀􀝔 converges\u003Cbr>Condition of Divergence:∫1∞ 􀝂 (􀝔)􀝀􀝔 diverges\u003Cbr>* Remainder: 0 \u003C 􀜴􀯇 ≤ ∫􀝂 (􀝔)􀝀􀝔 | 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>Root Test\u003Cbr>Series: ∑=1 􀜽􀯡\u003Cbr>Condition of Convergence: l􀯡 􀳙√|􀜽􀯡 | \u003C 1\u003Cbr>Condition of Divergence: lim 􀳙√|􀜽􀯡 | > 1 􀯡→∞\u003Cbr>* Test inconclusive ifl􀯡 􀳙√ |􀜽􀯡 | = 1\u003Cbr>| 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>Limit Comparison Test\u003Cbr>({􀜽􀯡 }, {􀜾􀯡 } > 0) Series: ∑=1 􀜽􀯡\u003Cbr>Condition of Convergence:\u003Cbr>􀜽􀯡 lim ~~ ~~ = 􀜮 > 0 􀯡→∞ 􀜾􀯡\u003Cbr>and ∑=0 􀜾􀯡 converges\u003Cbr>Condition of Divergence: lim 􀜽􀯡 = 􀜮 > 0 􀯡→∞ 􀜾􀯡\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 + ⋯ ) |","cbCailAbpsraWyPS","https://ap.wps.com/l/cbCailAbpsraWyPS","pdf",487675,"English","# Series Convergence Tests\n## 1-3: Divergence, Geometric, p-Series\n## 4-6: Alternating, Integral, Ratio\n## 7-10: Root, Comparison, Telescoping","[{\"question\":\"What does the nth-term test determine for a series?\",\"answer\":\"If the nth term does not go to zero (an ≠ 0), the series diverges. If it does go to zero, the test is inconclusive and cannot confirm convergence.\"},{\"question\":\"When can the geometric series be shown to converge?\",\"answer\":\"A geometric series converges when the absolute value of the common ratio is less than 1, i.e., |r| \\u003c 1.\"},{\"question\":\"How does the ratio test decide convergence or divergence?\",\"answer\":\"Compute lim n→∞ |a(n+1)|/|a(n)|. If the limit is less than 1, the series converges; if greater than 1, it diverges; if equal to 1, the test is inconclusive.\"}]","Cheat Sheet - Math Series Convergence Tests - 1-10 | PDF",1791925772]