[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-311543-105":59,"doc-detail-311543-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","convergence-tests-for-calculus-bc","Convergence Tests for Calculus BC","","Reference chart of key infinite series convergence tests used on the Calculus BC exam. Covers nth-term, geometric series, alternating series, integral, p-series, direct comparison, limit comparison, absolute vs conditional convergence, and ratio test, including the stated convergence or divergence conditions and brief notes on limitations such as when a converse is false or a test is not applicable. Also advises using another test if one is inconclusive.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/convergence-tests-for-calculus-bc/311543/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/convergence-tests-for-calculus-bc/311543.png","ImageObject",300,407,{"name":92,"@type":93},"Xiajie","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-09-20",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",11,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"Which convergence tests are included for Calculus BC practice?","Question",{"text":113,"@type":114},"The chart includes the nth-term test, geometric series, alternating series test, integral test, p-series, direct comparison test, limit comparison test, absolute convergence implies convergence, and ratio test.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What does the chart recommend if a test is inconclusive?",{"text":118,"@type":114},"It advises trying a different convergence test, because no single test must be used for every series.",{"name":120,"@type":111,"acceptedAnswer":121},"Are there limitations on using certain tests?",{"text":122,"@type":114},"Yes. For example, the nth-term test includes a note about divergence-only use, and the ratio test cannot be used when the relevant limit condition yields an inapplicable form.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},311543,1790531674,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":8,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":81,"preview_page_count":4},8814010472675,"https://avatar.qwps.com/avatar/WGlhamll","Summary of the convergence tests that may appear on the Calculus BC exam.  \n\n| Test Name | The series … | will converge if |  |  |  | Or will diverge if | Comments |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| nth –term test |  an\u003Cbr>n 􀀠 1 |  |  |  |  | ln an 􀁺 0 | For divergence only; the converse is false. |\n| Geometric series |  arn􀀐1 n 􀀠 1 | 􀀐1 􀀟 r 􀀟 1 |  |  |  | r 􀁤 􀀐 1 or r 􀁴 1 | a Sum = ~~ ~~\u003Cbr>1 􀀐 r |\n| Alternating series test |  (􀀐1)n􀀐1ann 􀀠 1 | an􀀎1  􀀟 an  and ln an 􀀠 0 |  |  |  |  | Error bound\u003Cbr>S􀁦 􀀐 Sn 􀀟 an􀀎1 |\n| Integral test |  an and\u003Cbr>n 􀀠 1\u003Cbr>an 􀀠 f 􀀋n 􀀌 􀁴 0 | 􀂳1􀁦 f 􀀋x 􀀌 dx converges |  |  |  | 􀂳1􀁦 f 􀀋x 􀀌 dx diverges | f must be\u003Cbr>continuous,\u003Cbr>positive and\u003Cbr>decreasing. |\n| p-series | 􀁦\u003Cbr>􀂦\u003Cbr>n 􀀠 1\u003Cbr>1\u003Cbr>np | p 􀀡 1 |  |  |  | p 􀁤 1 |  |\n| Direct comparison test |  an\u003Cbr>n 􀀠 1 | 0 􀀟 an 􀁤 bn and |  |  |  bn\u003Cbr>n 􀀠 1 | 0 􀀟 bn 􀁤 an and  bn diverges\u003Cbr>n 􀀠 1 |  |\n|  |  | converges |  |  |  |  |  |\n| Limit comparison test |  an\u003Cbr>n 􀀠 1 | an 􀀡 0, bn 􀀡 0 lim  an  􀀠 L 􀀡 0 n􀁯􀁦 bn\u003Cbr>and\u003Cbr> bn\u003Cbr>n 􀀠 1\u003Cbr>converges |  |  |  | an 􀀡 0, bn 􀀡 0 lim  an  􀀠 L 􀀡 0 n􀁯􀁦 bn\u003Cbr>and  bn\u003Cbr>n 􀀠 1\u003Cbr>diverges |  |\n| Absolute convergence implies convergence |  an\u003Cbr>n 􀀠 1 | 􀁦\u003Cbr>􀂦\u003Cbr>n 􀀠 1 | an | converges |  | If  an  diverges,  an may still\u003Cbr>n 􀀠 1 n 􀀠 1\u003Cbr>converge. (Conditional convergence) . |  |\n| Ratio test |  an\u003Cbr>n 􀀠 1 | lim  | an􀀎1 | 􀀟 1 n􀁯􀁦 | an | |  |  |  | lim  | an􀀎1 | 􀀡 1 n􀁯􀁦 | an | | If lim  | an􀀎1 | 􀀠 1\u003Cbr>n􀁯􀁦 | an | the ratio test cannot be used. |\n\nAnother useful convergence tests that may be used, but not tested:  \n\n| Test Name | The series … | will converge if | Or will diverge if |  | Comments |\n| --- | --- | --- | --- | --- | --- |\n| Root Test |  an\u003Cbr>n 􀀠 1 | ln nan  􀀟 1\u003Cbr>|  | ln nan  􀀡 1\u003Cbr>| The test cannot  be used if   ln nan  􀀠 1 \u003Cbr>|\n\nIf one test is inconclusive, try a different test. There is no single test that must be used on a given series.","cbCaiptKkSmOghAQ","https://ap.wps.com/l/cbCaiptKkSmOghAQ","pdf",771231,"English","# Convergence Tests for Calculus BC\n## Tested Series Convergence Tests\n## Additional Uncovered Test Notes","[{\"question\":\"Which convergence tests are included for Calculus BC practice?\",\"answer\":\"The chart includes the nth-term test, geometric series, alternating series test, integral test, p-series, direct comparison test, limit comparison test, absolute convergence implies convergence, and ratio test.\"},{\"question\":\"What does the chart recommend if a test is inconclusive?\",\"answer\":\"It advises trying a different convergence test, because no single test must be used for every series.\"},{\"question\":\"Are there limitations on using certain tests?\",\"answer\":\"Yes. For example, the nth-term test includes a note about divergence-only use, and the ratio test cannot be used when the relevant limit condition yields an inapplicable form.\"}]","Convergence Tests for Calculus BC | PDF",1789894432]