[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-572873-105":3,"detail-sidebar-cat-1-en-105":80,"doc-detail-572873-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","calculus-ii-series-things-to-consider-convergence-and-divergence-cheat-sheet","Calculus II - Series - Things to Consider - Convergence and Divergence Cheat Sheet","","Calculus II cheat sheet focused on convergence and divergence of infinite series. Presents practical guidance on selecting appropriate tests and emphasizes starting with the Test for Divergence when possible. Summarizes multiple standard criteria, including P-Series, Geometric Series, Direct and Limit Comparison, Root Test, Alternating Series Test, Ratio Test, and Integral Test, with key structural forms to recognize each case. Includes brief reminders that these are suggestions rather than guaranteed rules.",{"@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/calculus-ii-series-things-to-consider-convergence-and-divergence-cheat-sheet/572873/",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/calculus-ii-series-things-to-consider-convergence-and-divergence-cheat-sheet/572873.png","ImageObject",442,249,{"name":42,"@type":43},"Oliver","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},"Why should you consider the Test for Divergence first when checking a series?","Question",{"text":62,"@type":63},"It allows concluding divergence by identifying a required condition, without additional work on other tests if the series fails the divergence criterion.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How do you recognize when the P-Series Test applies?",{"text":67,"@type":63},"Use the form of a p-series as indicated in the sheet, where the general term matches a power-law pattern dependent on n.",{"name":69,"@type":60,"acceptedAnswer":70},"When can the Integral Test be used for a series?",{"text":71,"@type":63},"When the related sequence is continuous, positive, and decreasing; the sheet also notes using a first-derivative check for decreasing behavior.","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},572873,1792403984,{"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},8796095461610,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Calculus II  \nSeries-Things to Consider  \nImportant: This cheat sheet is not intended to be a list of guaranteed rules to follow. This introduces some hints and some ideas you may consider when choosing tests for convergence or divergence when evaluating a given series.  \nIt is usually a good idea to try using the Test for Divergence as a first step when evaluating a series for convergence or divergence. If we can show that:  \nnlan  0  \nThen we can say that the series diverges without having to do any extra work.  \nBelow are some general cases in which each test may help:  \nP-Series Test:  \n• The series be written in the form: P n~~1p~~  \nGeometric Series Test:  \n• When the series can be written in the form: Pan rn−1 or Pan rn  \nDirect Comparison Test:  \n• When the given series, an looks like a known, or more simple, series, bn  \nLimit Comparison Test:  \n• When you can see that the series looks like another convergent or divergent series, bn  \n• But it is hard to say whether bn > an or bn \u003C an  \nRoot Test:  \n• When the series can be written in the form: P (an)n  \nAlternating Series Test:  \n• When the series can be written in the form: P (−1)n+1an or P (−1)nan  \nRatio Test:  \n• Whenever we are given something involving a factorial, e.g. n!  \n• Whenever we are given something involving a constant raised to the nth power, e.g. P ~~n~~5~~+n5~~  \nIntegral Test:  \n• If the sequence is:  \n– continuous  \n– positive  \n– decreasing (we can use the First Derivative Test here)  \n [csusm.edu/stemsc](csusm.edu/stemsc)  xxx  @csusm_stemcenter Tel:  \nNorth: 760-750-4101  \nSouth: 760-750-7324  \nCalculus II  \nSeries-Things to Consider  \nRemember: These are just suggestions. There are other tests which may get us to the same answer.  \n| Convergent |  | Divergent |  |\n| --- | --- | --- | --- |\n|  n2n2~~ ~~1n |  |  11 |  |\n|  153n3 |  | ∞X (~~ ~~−323)nn+1 n=1 |  |\n| ∞X (~~ ~~−323)nn+1 n=1 |  |  n2n2~~ ~~1n |  |\n| ∞X tann1~~n~~(n) n=1 |  |  n2n2~~ ~~1n |  |\n|  n2~~ ~~~~ ~~n |  | ∞ X\u003Cbr>n=1 | n2 − 1\u003Cbr>n3 + 4 |\n| ∞ X\u003Cbr>n=1 | 􀀐 ~~ ~~1~~ ~~n8n 􀀑 n |  (nn4!)nn |  |\n| ∞X(−1)n n2n2~~ ~~1nn=1 |  | ∞X(−1)n ~~ ~~11 n=1 |  |\n| ∞X (n2n2!~~ ~~)! n=1 |  |  11 |  |\n|  en1n2 |  |  11 |  |\n\n[csusm.edu/stemsc](csusm.edu/stemsc)  \nxxx  \n @csusm_stemcenter Tel:  \nNorth: 760-750-4101  \nSouth: 760-750-7324","cbCaibTzAPT7Zdwt","https://ap.wps.com/l/cbCaibTzAPT7Zdwt","pdf",284481,"English","# Choosing Series Convergence/Divergence Tests\n## Test for Divergence First\n## Common Test Cases\n### P-Series Test\n### Geometric Series Test\n### Direct Comparison Test\n### Limit Comparison Test\n### Root Test\n### Alternating Series Test\n### Ratio Test\n### Integral Test","[{\"question\":\"Why should you consider the Test for Divergence first when checking a series?\",\"answer\":\"It allows concluding divergence by identifying a required condition, without additional work on other tests if the series fails the divergence criterion.\"},{\"question\":\"How do you recognize when the P-Series Test applies?\",\"answer\":\"Use the form of a p-series as indicated in the sheet, where the general term matches a power-law pattern dependent on n.\"},{\"question\":\"When can the Integral Test be used for a series?\",\"answer\":\"When the related sequence is continuous, positive, and decreasing; the sheet also notes using a first-derivative check for decreasing behavior.\"}]","Calculus II - Series - Things to Consider - Convergence and Divergence Cheat Sheet | PDF",1791929808]