[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572086-105":53,"doc-detail-572086-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","sums-and-asymptotics-chapter-14-introduction","Sums and Asymptotics - Chapter 14 - Introduction","","Counting and closed-form expressions underpin the analysis of algorithms, probabilistic systems, and practical verification of counting arguments. The material motivates counting beyond simple cases, introducing core proof techniques and the need to confirm correctness via alternative counts or bijections. It then presents Chapter 14, focused on sums and asymptotic behavior: closed forms, verification by induction, geometric and annuity-motivated examples, and the role of asymptotic notation for describing growth and bounding error terms when exact formulas are unavailable.",{"@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":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/sums-and-asymptotics-chapter-14-introduction/572086/",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/sums-and-asymptotics-chapter-14-introduction/572086.png","ImageObject",442,249,{"name":88,"@type":89},"LangkahRina","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-22","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":73},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"Why is counting important in computer science?","Question",{"text":108,"@type":109},"Counting helps determine time and storage requirements for computational problems, underlies probability theory, and supports proof techniques such as the pigeonhole principle and combinatorial proofs.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What is the goal of Chapter 15 versus Chapter 14 in this excerpt?",{"text":113,"@type":109},"Chapter 15 focuses on basic rules for determining the cardinality of a set, emphasizing simple counting as a practical skill. Chapter 14 starts studying sums and asymptotics and how to derive closed-form expressions.",{"name":115,"@type":106,"acceptedAnswer":116},"What do “closed forms” and asymptotic notation provide?",{"text":117,"@type":109},"Closed forms express sums or products without explicit subscripted summations/products, making growth and limiting behavior clearer. Asymptotic notation is used to bound error terms and describe the order of magnitude when exact closed forms are unavailable.","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},572086,1792669279,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":73,"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":141},962090893776,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","“mcs”— 2012/1/4 — 13:53 — page 417 — \\#425  \nIII Counting  \n“mcs”— 2012/1/4 — 13:53 — page 418 — \\#426  \n“mcs”— 2012/1/4 — 13:53 — page 419 — \\#427  \nIntroduction  \nCounting is useful in computer science for several reasons:  \n􀀏 Determining the time and storage required to solve a computational problem  \n—a central objective in computer science—often comes down to solving a counting problem.  \n􀀏 Counting is the basis of probability theory, which plays a central role in all sciences, including computer science.  \n􀀏 Two remarkable proof techniques, the “pigeonhole principle” and “combinatorial proof,” rely on counting.  \nCounting seems easy enough: 1, 2, 3, 4, etc. This direct approach works well for counting simple things—like your toes—and may be the only approach for extremely complicated things with no identifiable structure. However, subtler methods can help you count many things in the vast middle ground, such as:  \n􀀏 The number of different ways to select a dozen doughnuts when there are five varieties available.  \n􀀏 The number of 16-bit numbers with exactly 4 ones.  \nPerhaps surprisingly, but certainly not coincidentally, these two numbersa are the same: 1820 .  \nWe begin our study of counting in Chapter 14 with a collection of rules and methods for finding closed-form expressions for commonly-occurring sums and products such as PniD 1 xi and n􀂊 D QniD 1 i. We also introduce asymptotic notations such as 􀀘 , O , and 􀂂 that are commonly used in computer science to express  \n“mcs”— 2012/1/4 — 13:53 — page 420 — \\#428  \n420 Part III Counting  \nthe how a quantity such as the running time of a program grows with the size of the input.  \nChapter 15 describes the most basic rules for determining the cardinality of a set. These rules are actually theorems, but our focus won't be on their proofs per se  \n—our objective is to teach you simple counting as a practical skill, like integration. But counting can be tricky, and people make counting mistakes all the time, so a crucial part of counting skill is being able to verify a counting argument.  \nSometimes this can be done simply by finding an alternative way to count and then comparing answers—they better agree. But most elementary counting arguments reduce to finding a bijection between objects to be counted and easy-to-count sequences. The chapter shows how explicitly defining these bijections—and verifying that they are bijections—is another useful way to verify counting arguments. The material in Chapter 15 is simple yet powerful, and it provides a great tool set for use in your future career.  \n“mcs”— 2012/1/4 — 13:53 — page 421 — \\#429  \n14 Sums and Asymptotics  \nSums and products arise regularly in the analysis of algorithms, financial applications, physical problems, and probabilistic systems. For example, according to Theorem 2.2.1,  \n1 C 2 C 3 C 􀀁 􀀁 􀀁 C n D n.n~~ ~~C2~~ ~~1/ : (14.1)  \nOf course the lefthand sum could be expressed concisely as a subscripted summation n  \nXi;  \ni D 1  \nbut the right hand expression n.n C 1/=2 is not only concise, it is also easier to evaluate, and it more clearly reveals properties such as the growth rate of the sum. Expressions like n.n C 1/=2 that do not make use of subscripted summations or products —or those handy but sometimes troublesome dots —are called closed forms.  \nAnother example is the closed form for a geometric sum  \n1 C x C x 2 C x 3 C 􀀁 􀀁 􀀁 C xn D 1~~ ~~~~ ~~nCx1~~ ~~ (14.2) given in Problem 6.2. The sum as described on the left hand side of (14.2) involves n additions and 1 C 2 C 􀀁 􀀁 􀀁 C .n 􀀀 1/ D .n 􀀀 1/n=2 multiplications, but its closed form on the right hand side can be evaluated using fast exponentiation with at most  \n2 log n multiplications and a couple of substractions. Also, the closed form makes the growth and limiting behavior of the sum much more apparent.  \nEquations (14.1) and (14.2) were easy to verify by induction, but, as is often the case, the proofs by induction gave no hint about how these formulas were foun","cbCaiksPBeEFA4eX","https://ap.wps.com/l/cbCaiksPBeEFA4eX","pdf",2650406,404,"English","# Introduction\n# 14 Sums and Asymptotics\n## 14.1 The Value of an Annuity","[{\"question\":\"Why is counting important in computer science?\",\"answer\":\"Counting helps determine time and storage requirements for computational problems, underlies probability theory, and supports proof techniques such as the pigeonhole principle and combinatorial proofs.\"},{\"question\":\"What is the goal of Chapter 15 versus Chapter 14 in this excerpt?\",\"answer\":\"Chapter 15 focuses on basic rules for determining the cardinality of a set, emphasizing simple counting as a practical skill. Chapter 14 starts studying sums and asymptotics and how to derive closed-form expressions.\"},{\"question\":\"What do “closed forms” and asymptotic notation provide?\",\"answer\":\"Closed forms express sums or products without explicit subscripted summations/products, making growth and limiting behavior clearer. Asymptotic notation is used to bound error terms and describe the order of magnitude when exact closed forms are unavailable.\"}]","Sums and Asymptotics - Chapter 14 - Introduction | PDF",1791925406,141]