[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572263-105":53,"doc-detail-572263-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","mathematics-cheat-sheet","Mathematics Cheat Sheet","","A compact mathematics reference compiling core definitions and formulaic tools used across topics. Includes asymptotic growth notations such as Ω/Θ/Ο, limit superior and inferior, and concepts of supremum/infimum. Provides standard series results like geometric and harmonic series relations, along with combinatorics and special number sequences including combinations and Stirling numbers, Eulerian numbers, and Catalan numbers. Also summarizes common recurrence techniques such as the Master method, substitution, summing factors, and generating functions steps.",{"@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/mathematics-cheat-sheet/572263/",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/mathematics-cheat-sheet/572263.png","ImageObject",442,249,{"name":88,"@type":89},"moss~","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-22","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":79},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What asymptotic notations and limit concepts are included?","Question",{"text":108,"@type":109},"The sheet lists Ω, Θ, O relationships and definitions for limit, lim inf, and lim sup. It also includes supremum and infimum characterizations.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"Which combinatorics tools and special sequences are covered?",{"text":113,"@type":109},"It summarizes combinations and introduces Stirling numbers of both kinds, Eulerian numbers, and Catalan numbers. It also presents multiple identities related to these quantities.",{"name":115,"@type":106,"acceptedAnswer":116},"What recurrence-solving methods does the cheat sheet provide?",{"text":117,"@type":109},"It includes the Master method cases, substitution via transforming recurrences to simpler closed forms, summing factors using telescoping expansions, and a generating-function workflow with steps to choose and solve for the function.","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},572263,1792122291,{"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":79,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":135,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":79},962090895452,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","| Mathematics Cheat Sheet |  |  |  |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| Definitions |  |  | Series |  |  |  |  |\n| 􀦦(􀦭) = 􀦕(􀦧(􀦭)) |  | iff ∃ positive 􀦣, 􀦭0 such that 0 ≤􀦦(􀦭) ≤ 􀦣􀦧(􀦭) ∀􀦭 ≥ 􀦭0 . |  􀦨 = 􀦭(􀦭~~ ~~~~ ~~1) ,  􀦨 2 = 􀦭(􀦭~~ ~~+~~ ~~1~~ ~~)6(2􀦭~~ ~~+~~ ~~1) ,  􀦨 3 = 􀦭~~ ~~2~~ ~~(􀦭~~ ~~~~ ~~1)2 .\u003Cbr>􀦨=1 􀦨=1 􀦨=1\u003Cbr>In general:\u003Cbr>∑􀦨1 􀦨 􀦬 = 􀦬~~ ~~1+~~ ~~1 [(􀦭 + 1)􀦬+1 − 1 − ∑􀦨1 ((􀦨 + 1)􀦬+1 − 􀦨 􀦬+1 − (􀦬 + 1)􀦨 􀦬 ) ] 􀦨11 􀦨 􀦬 = 􀦬~~ ~~1+~~ ~~1 􀦪0 ( 􀦬  1 ) 􀦈􀦪􀦭􀦬+1−􀦪 .\u003Cbr>Geometric series: |  |  |  |  |\n| 􀦦(􀦭) = Ω(􀦧(􀦭)) |  | iff ∃ positive 􀦣, 􀦭0 such that 􀦦(􀦭) ≥􀦣􀦧(􀦭) ≥ 0 ∀􀦭 ≥ 􀦭0 . |  |  |  |  |  |\n| 􀦦(􀦭) = Θ(􀦧(􀦭)) |  | iff 􀦦(􀦭) = 􀦕(􀦧(􀦭)) and 􀦦(􀦭) =Ω(􀦧(􀦭)) . |  |  |  |  |  |\n| 􀦦(􀦭) = 􀦮(􀦧(􀦭)) |  | iff lim􀦭→∞ 􀦦(􀦭)/􀦧(􀦭) = 0 . |  |  |  |  |  |\n| lim 􀦭→∞ | 􀦡 = 􀦡􀦭 | iff ∀􀪘 > 0, ∃􀦭0 such that |􀦡 􀦭 −􀦡| \u003C 􀪘 , ∀􀦭 ≥ 􀦭0 . |  |  |  |  |  |\n| sup 􀦙 |  | least 􀦢 ∈ ℝ such that 􀦢 ≥ 􀦲 , ∀􀦲 ∈􀦙 . | 􀦭∑ 􀦣 􀦨 =􀦨=0\u003Cbr>􀦭∑ 􀦨􀦣 􀦨 =􀦨=0 | 􀦣 􀦭+1 − 1 ∞ 1\u003Cbr> 􀦣 − 1 , 􀦣 ≠ 1, ∑ 􀦣 􀦨 = 1 − 􀦣 ,􀦨=0\u003Cbr>􀦭􀦣 􀦭+2 − (􀦭 + 1)􀦣 􀦭+1 + 􀦣\u003Cbr>(􀦣 − 1)2 , 􀦣 ≠ 1, |  􀦣 􀦨 =􀦨=1 |  | 1~~ ~~~~ ~~􀦣 , |􀦣| \u003C 1, |\n| inf 􀦙 |  | greatest 􀦢 ∈ ℝ such that 􀦢 ≤ 􀦲 ,∀􀦲 ∈ 􀦙 . |  |  |  􀦨􀦣 􀦨 =􀦨=0 |  􀦣  , |􀦣| \u003C 1 .(1 − 􀦣)2 |  |\n| lim inf 􀦡􀦭\u003Cbr>􀦭→∞ |  | lim inf{􀦡 􀦨 ∣ 􀦨 ≥ 􀦭, 􀦨 ∈ ℕ} .􀦭→∞ | Harmonic series:\u003Cbr>􀦎􀦭 =  1􀦨 ,  􀦨􀦎􀦨 = 􀦭(􀦭~~ ~~~~ ~~1) 􀦎􀦭 − 􀦭(􀦭~~ ~~~~ ~~1) .\u003Cbr>􀦨=1 􀦨=1\u003Cbr>∑􀦨1 􀦎􀦨 = (􀦭 + 1)􀦎􀦭 − 􀦭, ∑􀦨1 ( 􀦨􀦬 ) 􀦎􀦨 = ( 􀦭􀦬 11 ) (􀦎􀦭+1 − 􀦬~~ ~~1+~~ ~~1 ) . |  |  |  |  |\n| lim sup 􀦡 􀦭􀦭→∞ |  | lim sup{􀦡 􀦨 ∣ 􀦨 ≥ 􀦭, 􀦨 ∈ ℕ} .􀦭→∞ |  |  |  |  |  |\n| ( 􀦭􀦪 ) |  | Combinations: Size 􀦪 subsets of a size 􀦭 set. |  |  |  |  |  |\n| [ 􀦭􀦪 ] |  | Stirling numbers (􀏭 st kind): Arrangements of an 􀦭 element set into 􀦪 cycles. | 1. ( 􀦭􀦪 ) = (􀦭~~ ~~−􀦭􀦪!)!􀦪! , 2. 􀦪0 ( 􀦭􀦪 ) = 2􀦭 , 3. ( 􀦭􀦪 ) = ( 􀦭  􀦪 ), 4. ( 􀦭􀦪 ) = 􀦭􀦪 ( 􀦭􀦪 − 11 ), 5. ( 􀦭􀦪 ) = ( 􀦭  1 ) + ( 􀦭􀦪 − 11 ), 6. ( 􀦭􀦬 ) ( 􀦬􀦪 ) = ( 􀦭􀦪 ) ( 􀦭􀦬 −􀦪􀦪 ), 7. 􀦪0 ( 􀦱  􀦪 ) = ( 􀦱 + 􀦭􀦭 + 1 ), 8. 􀦪0 ( 􀦪􀦬 ) = ( 􀦭􀦬  11 ), 9. 􀦪0 ( 􀦱􀦪 ) ( 􀦭  􀦪 ) = ( 􀦱  􀦲 ), 10. ( 􀦭􀦪 ) = (−1)􀦪 ( 􀦪 − 􀦭􀦪 − 1 ), 11. [ 􀦭1 ] = [ 􀦭􀦭 ] = 1, 12. [ 􀦭2 ] = 2􀦭−1 − 1, 13. [ 􀦭􀦪 ] = 􀦪 [ 􀦭  1 ] + [ 􀦭􀦪 − 11 ], |  |  |  |  |\n| [ 􀦭􀦪 ] |  | Stirling numbers (􀎆 nd kind): Partitions of an 􀦭 element set into 􀦪 non-empty sets. |  |  |  |  |  |\n| ⟨ 􀦭􀦪 ⟩ |  | 􀏭 st order Eulerian numbers: Permutations 􀪍1 􀪍2…􀪍􀦭 on {1, 2,…, 􀦭} with 􀦪 ascents. |  |  |  |  |  |\n| ⟪ 􀦭􀦪 ⟫ |  | 􀎆nd order Eulerian numbers. |  |  |  |  |  |\n| 􀦉􀦭 |  | Catalan Numbers: Binary trees with 􀦭 + 1 vertices. |  |  |  |  |  |\n| 14. [ 􀦭1 ] = (􀦭 − 1)!, 15. [ 􀦭2 ] = (􀦭 − 1)!􀦎􀦭−1 , 16. [ 􀦭􀦭 ] = 1, 17. [ 􀦭􀦪 ] ≥ [ 􀦭􀦪 ], 18. [ 􀦭􀦪 ] = (􀦭 − 1) [ 􀦭  1 ] + [ 􀦭􀦪 − 11 ], 19. [ 􀦭  1 ] = [ 􀦭  1 ] = ( 􀦭2 ), 20. 􀦪0 [ 􀦭􀦪 ] = 􀦭!, 21. 􀦉􀦭 = 􀦭~~ ~~+1~~ ~~1 ( 2􀦭􀦭 ), 22. ⟨ 􀦭0 ⟩ = ⟨ 􀦭  1 ⟩ = 1, 23. ⟨ 􀦭􀦪 ⟩ = ⟨ 􀦭 − 􀦭1 − 􀦪 ⟩, 24. ⟨ 􀦭􀦪 ⟩ = (􀦪 + 1) ⟨ 􀦭  1 ⟩ + (􀦭 − 􀦪) ⟨ 􀦭􀦪 − 11 ⟩, 25. ⟨ 0􀦪 ⟩ = { 10 otifhekri􀎅se, ,\u003Cbr>26. ⟨ 􀦭1 ⟩ = 2􀦭 − 􀦭 − 1, 27. ⟨ 􀦭2 ⟩ = 3􀦭 − (􀦭 + 1)2􀦭 + ( 􀦭 +2 1 ), 28. 􀦷􀦭 = 􀦪0 ⟨ 􀦭􀦪 ⟩ ( 􀦷  􀦪 ), 29. ⟨ 􀦭􀦬 ⟩ = 􀦪0 ( 􀦭  1 ) (􀦬 + 1 − 􀦪)􀦭 (−1)􀦪 , 30. 􀦬! [ 􀦭􀦬 ] = 􀦪0 ⟨ 􀦭􀦪 ⟩ ( 􀦭  􀦬 ), 31. ⟨ 􀦭􀦬 ⟩ = 􀦪0 [ 􀦭􀦪 ]( 􀦭  􀦪 ) (−1)􀦭−􀦪−􀦬 􀦪!, 32. ⟪ 􀦭0 ⟫ = 1, 33. ⟪ 􀦭􀦭 ⟫ = 0 for 􀦭 ≠ 0 , 34. ⟪ 􀦭􀦪 ⟫ = (􀦪 + 1) ⟪ 􀦭  1 ⟫ + (2􀦭 − 1 − 􀦪) ⟪ 􀦭􀦪 − 11 ⟫, 35. 􀦪0 ⟪ 􀦭􀦪 ⟫ = (22􀦭􀦭)􀦭~~ ~~ , 36. [ 􀦷  􀦭 ] = 􀦪0 ⟪ 􀦭􀦪 ⟫ ( 􀦷 + 􀦭  1 − 􀦪 ), 37. [ 􀦭􀦬 11 ] =  ( 􀦭􀦪 ) [ 􀦪􀦬 ] = 􀦪0 [ 􀦪􀦬 ](􀦬 + 1)􀦭−􀦪 , |  |  |  |  |  |  |  |\n\n􀏭  \n\n| Mathematics Cheat Sheet |  |\n| --- | --- |\n| Identities Cont. | Trees |\n| 38. [ 􀦭􀦬  11 ] =  [ 􀦭􀦪 ]( 􀦪􀦬 ) = 􀦪0 [ 􀦪􀦬 ] 􀦭􀦭−􀦪 = 􀦭! 􀦪0 􀦪1! [ 􀦪􀦬 ], 39. [ 􀦷  􀦭 ] = 􀦪0 ⟪ 􀦭􀦪 ⟫ ( 􀦷 􀦪 ), 40. [ 􀦭􀦬 ] =  ( 􀦭􀦪 ) [ 􀦪􀦬 11 ](−1)􀦭−􀦪 , 41. [ 􀦭􀦬 ] =  [ 􀦭􀦪  11 ]( 􀦪􀦬 ) (−1)􀦬−􀦪 , 42. [ 􀦬 +􀦭􀦬 + 1 ] = 􀦪0 􀦪 [ 􀦭  􀦪 ], 43. [ 􀦬 +􀦭􀦬 + 1 ] = 􀦪0 􀦪(􀦭 + 􀦪) [ 􀦭  􀦪 ], 44. ( 􀦭􀦬 ) =  [ 􀦭􀦪  11 ][ 􀦪􀦬 ](−1)􀦬−􀦪 , 45. (􀦭 − 􀦬)! ( 􀦭􀦬 ) =  [ 􀦭􀦪  11 ][ 􀦪􀦬 ](−1)􀦬−􀦪 , for 􀦭 ≥ 􀦬 , 46. [ 􀦭  􀦬 ] =  ( 􀦬􀦬  􀦭􀦪 ) ( 􀦬􀦭  􀦭􀦪 ) [ 􀦬  􀦪 ], 47. [ 􀦭  􀦬 ] =  ( 􀦬􀦬  􀦭􀦪 ) ( 􀦬􀦭  􀦭􀦪 ) [ 􀦬  􀦪 ], 48. [ ℓ  􀦬 ]( ℓ  􀦬 ) =  [ 􀦪ℓ ][ 􀦭  􀦪 ]( 􀦭􀦪 ), 49. [ ℓ  􀦬 ]( ℓ  􀦬 ) =  [ 􀦪ℓ ][ 􀦭  􀦪 ]( 􀦭􀦪 ) . | Every tree with 􀦭 vertices has 􀦭 − 1 edges.\u003Cbr>Kraft inequality: If the depths of ","cbCaigj2WSS8Q1cq","https://ap.wps.com/l/cbCaigj2WSS8Q1cq","pdf",114979,10,"English","# Definitions\n# Series and Limits\n# Combinations and Number Sequences\n# Identities and Trees\n# Recurrences\n## Master method\n## Substitution\n## Summing factors\n## Generating functions","[{\"question\":\"What asymptotic notations and limit concepts are included?\",\"answer\":\"The sheet lists Ω, Θ, O relationships and definitions for limit, lim inf, and lim sup. It also includes supremum and infimum characterizations.\"},{\"question\":\"Which combinatorics tools and special sequences are covered?\",\"answer\":\"It summarizes combinations and introduces Stirling numbers of both kinds, Eulerian numbers, and Catalan numbers. It also presents multiple identities related to these quantities.\"},{\"question\":\"What recurrence-solving methods does the cheat sheet provide?\",\"answer\":\"It includes the Master method cases, substitution via transforming recurrences to simpler closed forms, summing factors using telescoping expansions, and a generating-function workflow with steps to choose and solve for the function.\"}]","Mathematics Cheat Sheet | PDF",1791926485]