[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-167357-en":3,"doc-seo-167357-105":30,"detail-sidebar-cat-0-en-105":89},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},167357,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",4,"Exam","Infinite Series - Closed Forms - One-Table Cheat Sheet","Cheat sheet compiling common infinite series and their closed-form sums in a compact one-table reference. Covers geometric and shifted geometric series, linear coefficient patterns, quadratic/cubic and higher coefficient patterns, plus standard Taylor and Maclaurin series. For each entry, provides the series form, corresponding sigma notation, the simplified closed form, and the convergence range. Useful for quickly identifying sums and verifying results in calculus and discrete math.","INFINITE SERIES −→ CLOSED FORMS  \nONE-TABLE CHEAT SHEET  \n\n| Infinite series | Sigma notation | Closed form | Range |\n| --- | --- | --- | --- |\n| A. Geometric and shifted geometric series |  |  |  |\n| 1 + x + x2 + x3 + · · · | ∞ Xxnn=0 | 1\u003Cbr>1 − x | |x| \u003C 1 |\n| x + x2 + x3 + x4 + · · · | ∞ Xxnn=1 | x\u003Cbr>1 − x | |x| \u003C 1 |\n| a + ar + ar2 + ar3 + · · · | ∞ X arn n=0 | a\u003Cbr>1 − r | |r| \u003C 1 |\n| a + aqx + aq2x2 + aq3x3 + · · · | ∞ Xa (qx)nn=0 | a\u003Cbr>1 − qx | |qx| \u003C 1 |\n| 1 + xk + x2k + x3k + · · · | ∞ Xxknn=0 | 1\u003Cbr>1 − xk | |x| \u003C 1 |\n| xm + xm+k + xm+2k + · · · | ∞ Xxkn+mn=0 | xm\u003Cbr>1 − xk | |x| \u003C 1 |\n| 1 + x2 + x4 + x6 + · · · | ∞ Xx2nn=0 | 1\u003Cbr>1 − x2 | |x| \u003C 1 |\n| x + x3 + x5 + x7 + · · · | ∞ Xx2n+1 n=0 | x\u003Cbr>1 − x2 | |x| \u003C 1 |\n| 1 − x2 + x4 − x6 + · · · | ∞ X (−1)nx2nn=0 | 1\u003Cbr>1 + x2 | |x| \u003C 1 |\n| x − x3 + x5 − x7 + · · · | ∞\u003Cbr>X (−1)nx2n+1 n=0 | x\u003Cbr>1 + x2 | |x| \u003C 1 |\n| B. Linear coefficient patterns |  |  |  |\n| 1 + 2x + 3x2 + 4x3 + · · · | ∞\u003Cbr>X (n + 1)xnn=0 | 1\u003Cbr>(1 − x)2 | |x| \u003C 1 |\n| x + 2x2 + 3x3 + 4x4 + · · · | ∞ Xnxnn=0 | x\u003Cbr>(1 − x)2 | |x| \u003C 1 |\n| b + (a + b)x + (2a + b)x2 + · · · | ∞\u003Cbr>X (an + b)xnn=0 | b + (a − b)x\u003Cbr>(1 − x)2 | |x| \u003C 1 |\n| 1 + 3x + 5x2 + 7x3 + · · · | ∞\u003Cbr>X (2n + 1)xnn=0 | 1 + x\u003Cbr>(1 − x)2 | |x| \u003C 1 |\n| 3 + 5x + 7x2 + 9x3 + · · · | ∞\u003Cbr>X (2n + 3)xnn=0 | 3 − x\u003Cbr>(1 − x)2 | |x| \u003C 1 |\n| 4 + 7x + 10x2 + 13x3 + · · · | ∞\u003Cbr>X (3n + 4)xnn=0 | 4 − x\u003Cbr>(1 − x)2 | |x| \u003C 1 |\n\nContinued on next page  \nINFINITE SERIES −→ CLOSED FORMS MathPro Cheat Sheet  \n\n| Infinite series | Sigma notation |  |  |  | Closed form | Range |\n| --- | --- | --- | --- | --- | --- | --- |\n| 1 − x + x2 − x3 + · · · | ∞ X (−1)nxnn=0 |  |  |  | 1\u003Cbr>1 + x | |x| \u003C 1 |\n| 1 − 2x + 3x2 − 4x3 + · · · | ∞\u003Cbr>X (−1)n (n + 1)xn\u003Cbr>n=0 |  |  |  | 1\u003Cbr>(1 + x)2 | |x| \u003C 1 |\n| x − 2x2 + 3x3 − 4x4 + · · · | ∞ X (−1)n+1nxnn=1 |  |  |  | x\u003Cbr>(1 + x)2 | |x| \u003C 1 |\n| b − (a + b)x + (2a + b)x2 − · · · | ∞\u003Cbr>X (−1)n (an + b)xn\u003Cbr>n=0 |  |  |  | b − (a − b)x\u003Cbr>(1 + x)2 | |x| \u003C 1 |\n| 1 − 3x + 5x2 − 7x3 + · · · | ∞\u003Cbr>X (−1)n (2n + 1)xn\u003Cbr>n=0 |  |  |  | 1 − x\u003Cbr>(1 + x)2 | |x| \u003C 1 |\n| C. Quadratic, cubic and higher coefficient patterns |  |  |  |  |  |  |\n| 1 + 3x + 6x2 + 10x3 + · · · | ∞ X 􀀀 nn=0 | +\u003Cbr>2 | 2􀀁 xn |  | 1\u003Cbr>(1 − x)3 | |x| \u003C 1 |\n| 2x2 + 6x3 + 12x4 + 20x5 + · · · | ∞\u003Cbr>Xn (n − 1)xnn=0 |  |  |  | 2x2\u003Cbr>(1 − x)3 | |x| \u003C 1 |\n| x + 4x2 + 9x3 + 16x4 + · · · | ∞ Xn2xnn=0 |  |  |  | x(1 + x)\u003Cbr>(1 − x)3 | |x| \u003C 1 |\n| 1 + 4x + 9x2 + 16x3 + · · · | ∞\u003Cbr>X (n + 1)2xnn=0 |  |  |  | 1 + x\u003Cbr>(1 − x)3 | |x| \u003C 1 |\n| x − 4x2 + 9x3 − 16x4 + · · · | ∞\u003Cbr>X (−1)n+1n2xnn=1 |  |  |  | x(1 − x)\u003Cbr>(1 + x)3 | |x| \u003C 1 |\n| 1 − 4x + 9x2 − 16x3 + · · · | ∞\u003Cbr>X (−1)n (n + 1)2xn\u003Cbr>n=0 |  |  |  | 1 − x\u003Cbr>(1 + x)3 | |x| \u003C 1 |\n| 6x3 + 24x4 + 60x5 + · · · | ∞\u003Cbr>Xn (n − 1)(n − 2)xn\u003Cbr>n=0 |  |  |  | 6x3\u003Cbr>(1 − x)4 | |x| \u003C 1 |\n| x + 8x2 + 27x3 + 64x4 + · · · | ∞ Xn3xnn=0 |  |  |  | x(1 + 4x + x2 )\u003Cbr>(1 − x)4 | |x| \u003C 1 |\n| 1 + 8x + 27x2 + 64x3 + · · · | ∞\u003Cbr>X (n + 1)3xnn=0 |  |  |  | 1 + 4x + x2\u003Cbr>(1 − x)4 | |x| \u003C 1 |\n| x + 16x2 + 81x3 + 256x4 + · · · | ∞ Xn4xnn=0 |  |  |  | x(1 + 11x + 11x2 + x3 )\u003Cbr>(1 − x)5 | |x| \u003C 1 |\n| Xn (n − 1) · · · (n − k + 1)xn | ∞ Xnk xnn=0 |  |  |  | k!xk\u003Cbr>(1 − x)k+1 | |x| \u003C 1 |\n| 1 + mx + 􀀀 m + 12􀀁 x2 + · · · | ∞ X 􀀀 nn=0 | + m −\u003Cbr>m − 1 |  | 1􀀁 xn | 1\u003Cbr>(1 − x)m | |x| \u003C 1 |\n| D. Standard Taylor and Maclaurin series |  |  |  |  |  |  |\n\nContinued on next page  \nINFINITE SERIES −→ CLOSED FORMS MathPro Cheat Sheet  \n\n| Infinite series | Sigma notation |  | Closed form | Range |\n| --- | --- | --- | --- | --- |\n| 1 + x + x22! + x33! + · · · | ∞\u003Cbr>X\u003Cbr>n=0 | xn\u003Cbr>n! | ex | x ∈ R |\n| 1 − x + x22! − x33! + · · · | ∞ X (−1n)n!~~ ~~xnn=0 |  | e − x | x ∈ R |\n| x − x33! + x55! − · · · | ∞\u003Cbr>X (~~ ~~−(12n)n2n1)+!~~ ~~1 n=0 |  | sin x | x ∈ R |\n| 1 − x22! + x44! − · · · | ∞ X (~~ ~~−~~ ~~1(2)nnx)!2nn=0 |  | cos x | x ∈ R |\n| x + x33! + x55! + · · · | ∞\u003Cbr>X (x22n11)! n=0 |  | sinhx | x ∈ R |\n| 1 + x22! + x44! + · · · | ∞ X (x","cbCaipjrUlDGjxSU","https://ap.wps.com/l/cbCaipjrUlDGjxSU","pdf",133720,1,3,"English","en",105,"# Geometric and shifted geometric series\n# Linear coefficient patterns\n# Quadratic, cubic and higher coefficient patterns\n# Standard Taylor and Maclaurin series","[{\"question\":\"What types of infinite series are included in the cheat sheet?\",\"answer\":\"It includes geometric and shifted geometric series, linear coefficient patterns, higher-order polynomial coefficient patterns, and standard Taylor/Maclaurin series.\"},{\"question\":\"Does the cheat sheet provide both series expressions and sigma notation?\",\"answer\":\"Yes. Each row lists the infinite series, its sigma notation, the closed-form sum, and the range of validity.\"},{\"question\":\"How can I use the convergence range when applying a closed form?\",\"answer\":\"Use the listed range (e.g., conditions like |x| \\u003c 1) to ensure the closed form applies for the given parameter values.\"}]","Infinite Series - Closed Forms - One-Table Cheat Sheet | PDF",1788212078,8,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":84,"head_meta":86,"extra_data":88,"updated_unix":28},"infinite-series-closed-forms-one-table-cheat-sheet","",{"@graph":36,"@context":83},[37,52,66],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":21},"https://docshare.wps.com/document/exam/",{"item":51,"name":13,"@type":43,"position":11},"https://docshare.wps.com/document/infinite-series-closed-forms-one-table-cheat-sheet/167357/",{"url":51,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":23,"description":14,"dateModified":60,"datePublished":60,"encodingFormat":59,"isAccessibleForFree":61,"interactionStatistic":62},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":41,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-08-31",true,{"@type":63,"interactionType":64,"userInteractionCount":4},"InteractionCounter",{"@type":65},"ViewAction",{"@type":67,"mainEntity":68},"FAQPage",[69,75,79],{"name":70,"@type":71,"acceptedAnswer":72},"What types of infinite series are included in the cheat sheet?","Question",{"text":73,"@type":74},"It includes geometric and shifted geometric series, linear coefficient patterns, higher-order polynomial coefficient patterns, and standard Taylor/Maclaurin series.","Answer",{"name":76,"@type":71,"acceptedAnswer":77},"Does the cheat sheet provide both series expressions and sigma notation?",{"text":78,"@type":74},"Yes. Each row lists the infinite series, its sigma notation, the closed-form sum, and the range of validity.",{"name":80,"@type":71,"acceptedAnswer":81},"How can I use the convergence range when applying a closed form?",{"text":82,"@type":74},"Use the listed range (e.g., conditions like |x| \u003C 1) to ensure the closed form applies for the given parameter values.","https://schema.org",{"og:url":51,"og:type":85,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":87,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":90},[91,95,99,102,107,112,117,121,126,129,133],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":92,"show_sort_weight":93,"slug":94},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":96,"show_sort_weight":97,"slug":98},"Literature",80,"literature",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":100,"slug":101},70,"exam",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},5,"Comic",60,"comic",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},6,"Technology",50,"technology",{"id":113,"doc_module":4,"doc_module_name":46,"category_name":114,"show_sort_weight":115,"slug":116},7,"Healthcare",40,"healthcare",{"id":29,"doc_module":4,"doc_module_name":46,"category_name":118,"show_sort_weight":119,"slug":120},"Research & Report",30,"research-report",{"id":122,"doc_module":4,"doc_module_name":46,"category_name":123,"show_sort_weight":124,"slug":125},9,"Religion & Spirituality",20,"religion-spirituality",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":127,"show_sort_weight":124,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":46,"category_name":135,"show_sort_weight":103,"slug":136},19,"General","general"]