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Series and Their Notations

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Cấu trúc

  • Series and Their Notations

  • Using Summation Notation

  • Using the Formula for Arithmetic Series

  • Using the Formula for Geometric Series

  • Using the Formula for the Sum of an Infinite Geometric Series

    • Determining Whether the Sum of an Infinite Geometric Series is Defined

    • Finding Sums of Infinite Series

  • Solving Annuity Problems

  • Key Equations

  • Key Concepts

  • Section Exercises

    • Verbal

    • Algebraic

    • Graphical

    • Numeric

    • Extensions

    • Real-World Applications

Nội dung

Chapter 2: Attackers and Their AttacksSecurity+ Guide to Network Security Fundamentals Second Edition Objectives•Develop attacker profiles•Describe basic attacks•Describe identity attacks•Identify denial of service attacks•Define malicious code (malware) Developing Attacker Profiles•Six categories: –Hackers–Crackers–Script kiddies–Spies–Employees–Cyberterrorists Developing Attacker Profiles (continued) Hackers•Person who uses advanced computer skills to attack computers, but not with a malicious intent•Use their skills to expose security flaws •Person who violates system security with malicious intent •Have advanced knowledge of computers and networks and the skills to exploit them•Destroy data, deny legitimate users of service, or otherwise cause serious problems on computers and networksCrackers •Break into computers to create damage•Are unskilled users•Download automated hacking software from Web sites and use it to break into computers•Tend to be young computer users with almost unlimited amounts of leisure time, which they can use to attack systemsScript Kiddies •Person hired to break into a computer and steal information•Do not randomly search for unsecured computers to attack•Hired to attack a specific computer that contains sensitive informationSpies •One of the largest information security threats to business•Employees break into their company’s computer for these reasons:–To show the company a weakness in their security–To say, “I’m smarter than all of you”–For moneyEmployees •Experts fear terrorists will attack the network and computer infrastructure to cause panic •Cyberterrorists’ motivation may be defined as ideology, or attacking for the sake of their principles or beliefs•One of the targets highest on the list of cyberterrorists is the Internet itselfCyberterrorists [...]...• Person hired to break into a computer and steal information • Do not randomly search for unsecured computers to attack • Hired to attack a specific computer that contains sensitive information Spies • Category of attacks in which the attacker attempts to assume the identity of a valid user Examining Identity Attacks • Similar to an active man-in-the-middle attack • Whereas... contents of a message before sending it on, a replay attack only captures the message and then sends it again later • Takes advantage of communications between a network device and a file server Replay Summary (continued) • Identity attacks attempt to assume the identity of a valid user • Denial of service (DoS) attacks flood a server or device with requests, making it unable to respond to valid... Identifying Denial of Service Attacks (continued) • Policies to minimize password-guessing attacks: – Passwords must have at least eight characters – Passwords must contain a combination of letters, numbers, and special characters – Passwords should expire at least every 30 days – Passwords cannot be reused for 12 months – The same password should not be duplicated and used on two or more systems Password... organization dropping below a specified level Password Guessing (continued) Objectives • Develop attacker profiles • Describe basic attacks • Describe identity attacks • Identify denial of service attacks • Define malicious code (malware) Identifying Denial of Service Attacks (continued) • Another DoS attack tricks computers into responding to a false request • An attacker can send a request to all... overwhelming it, and causing the server to crash or be unavailable to legitimate users • Encryption: changing the original text to a secret message using cryptography • Success of cryptography depends on the process used to encrypt and decrypt messages • Process is based on algorithms Weak Keys (continued) Summary • Six categories of Series and Their Notations Series and Their Notations By: OpenStaxCollege A couple decides to start a college fund for their daughter They plan to invest $50 in the fund each month The fund pays 6% annual interest, compounded monthly How much money will they have saved when their daughter is ready to start college in years? In this section, we will learn how to answer this question To so, we need to consider the amount of money invested and the amount of interest earned Using Summation Notation To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly The sum of the terms of a sequence is called a series Consider, for example, the following series + + 11 + 15 + 19 + The nth partial sum of a series is the sum of a finite number of consecutive terms beginning with the first term The notation Sn represents the partial sum S1 = S2 = + = 10 S3 = + + 11 = 21 S4 = + + 11 + 15 = 36 Summation notation is used to represent series Summation notation is often known as sigma notation because it uses the Greek capital letter sigma, Σ, to represent the sum Summation notation includes an explicit formula and specifies the first and last terms in the series An explicit formula for each term of the series is given to the right of the sigma A variable called the index of summation is written below the sigma The index of summation is set equal to the lower limit of summation, which is the number used to generate the first term in the series The number above the sigma, called the upper limit of summation, is the number used to generate the last term in a series 1/25 Series and Their Notations If we interpret the given notation, we see that it asks us to find the sum of the terms in the series ak = 2k for k = through k = We can begin by substituting the terms for k and listing out the terms of this series a1 = 2(1) = a2 = 2(2) = a3 = 2(3) = a4 = 2(4) = a5 = 2(5) = 10 We can find the sum of the series by adding the terms: ∑ 2k = + + + + 10 = 30 k=1 A General Note Summation Notation The sum of the firstnterms of a series can be expressed in summation notation as follows: n ∑a k k=1 This notation tells us to find the sum of ak from k = to k = n k is called the index of summation, is the lower limit of summation, and n is the upper limit of summation Q&A Does the lower limit of summation have to be 1? 2/25 Series and Their Notations No The lower limit of summation can be any number, but is frequently used We will look at examples with lower limits of summation other than How To Given summation notation for a series, evaluate the value Identify the lower limit of summation Identify the upper limit of summation Substitute each value of k from the lower limit to the upper limit into the formula Add to find the sum Using Summation Notation Evaluate ∑k k=3 According to the notation, the lower limit of summation is and the upper limit is So we need to find the sum of k2 from k = to k = We find the terms of the series by substituting k = 3,4,5,6, and into the function k2 We add the terms to find the sum ∑k = 32 + 42 + 52 + 62 + 72 k=3 = + 16 + 25 + 36 + 49 = 135 Try It Evaluate ∑ (3k – 1) k=2 38 Using the Formula for Arithmetic Series Just as we studied special types of sequences, we will look at special types of series Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference,d The sum of the terms of an arithmetic sequence is called an arithmetic series We can write the sum of the first n terms of an arithmetic series as: 3/25 Series and Their Notations Sn = a1 + (a1 + d) + (a1 + 2d) + + (an – d) + an We can also reverse the order of the terms and write the sum as Sn = an + (an – d) + (an – 2d) + + (a1 + d) + a1 If we add these two expressions for the sum of the first nterms of an arithmetic series, we can derive a formula for the sum of the first n terms of any arithmetic series Sn = a1 + (a1 + d) + (a1 + 2d) + + (an – d) + an + Sn = an + (an – d) + (an – 2d) + + (a1 + d) + a1 2Sn = (a1 + an) + (a1 + an) + + (a1 + an) Because there are n terms in the series, we can simplify this sum to 2Sn = n(a1 + an) We divide by to find the formula for the sum of the first n terms of an arithmetic series n(a1 + an) A General Note Formula for the Sum of the First n Terms of an Arithmetic Series Sn = An arithmetic series is the sum of the terms of an arithmetic sequence The formula for the sum of the first n terms of an arithmetic sequence is n(a1 + an) How To Sn = Given terms of an arithmetic series, find the sum of the first n terms Identify a1 and an Determine n Substitute values for a1, an, and n into the formula Sn = Simplify to find Sn Finding the First n Terms of an Arithmetic Series n(a1 + an) Find the sum of each arithmetic series + + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32 4/25 Series and Their Notations 20 + 15 + 10 +…+ −50 12 ∑ 3k − k=1 We are given ...Int. J. Med. Sci. 2010, 7 http://www.medsci.org 272IInntteerrnnaattiioonnaall JJoouurrnnaall ooff MMeeddiiccaall SScciieenncceess 2010; 7(5):272-277 © Ivyspring International Publisher. All rights reserved Research Paper Predictors of hepatic steatosis in HBeAg-negative chronic hepatitis B patients and their diagnostic values in hepatic fibrosis Rui-dan Zheng1, Cheng-run Xu1, Li Jiang1, Ai-xia Dou 2, Kun Zhou 2, Lun-gen Lu2  1. Research and Therapy Center for Liver Diseases, Southeast Hospital, Zhangzhou 363000, China 2. Department of Gastroenterology, Shanghai First People’s Hospital, Shanghai Jiaotong University School of Medicine, Shanghai 200080, China  Corresponding author: Lun-Gen Lu, M.D., Department of Gastroenterology, Shanghai First People’s Hospital, Shanghai Jiaotong University School of Medicine, Shanghai 200080, China. Tel: +86-21-63240090; Fax: +86-21-63241377; E-mail:live.0000@live.cn Received: 2010.05.05; Accepted: 2010.08.08; Published: 2010.08.11 Abstract Objective: To investigate predictors of hepatic steatosis in HBeAg-negative chronic hepatitis B (CHB) patients and their diagnostic values in hepatic inflammation and fibrosis. Methods: A total of 106 HBeAg-negative CHB patients with clinically and pathologically proven steatosis and 98 patients without steatosis were recruited into this study. The levels of fasting blood glucose (FBG), fasting insulin (FINS), triglyceride (TG), cholesterol (CHOL), alanine amino-transferase (ALT), aspartate aminotransferase (AST), albumin (Alb), globulin (Glb), HBV DNA, body mass index (BMI), homeostatic model assessment of insulin resistance (HOMA-IR) and pathological changes of the liver in inflammation, fibrosis and fatty deposition were examined in all patients. Results: The levels of BMI, HOMA-IR, FBG, insulin, TG, and CHOL were significantly higher in patients with steatosis than those without steatosis (all P<0.05). But ALT, AST and HBV DNA levels were significantly lower in patients with steatosis (all P<0.05). Logistic regression analysis showed that only FINS was a significant predictor for hepatic steatosis (P<0.05); FINS and Glb were significant predictors for hepatic inflammation (all P<0.05); BMI and TC were significant predictors for hepatic fibrosis (all P<0.05). Con-clusions: Hepatic steatosis, a common disease in HBeAg-negative CHB patients, was posi-tively associated with BMI, FBG, FINS, TG, TC, GGT, ALP and HOMA-IR. In these patients, the prevalence of hepatic inflammation and fibrosis was also increased. Key words: HBeAg negative, chronic hepatitis B, nonalcoholic fatty liver disease, liver biopsy Introduction The prevalence of HBeAg-negative chronic he-patitis B (CHB) tends to increase in recent years (1). With the improvement of living standard and nutri-tion status, hepatic steatosis frequently occurs in CHB patients. It has been shown that the incidence of he-patic steatosis in CHB patients was about 32% (2). The distribution of hepatic triglyceride content (HTGC) in 2, 287 subjects from a multiethnic, population-based sample (32.1% white, 48.3% black, and 17.5% His-panic) was previously examined and compared using proton magnetic resonance spectroscopy. Almost one third of the population had hepatic steatosis, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LEAN MANUFACTURING PRINCIPLES AND THEIR APPLICATION Mehmet C Kocakülâh; Jason F Brown; Joshua W Thomson Cost Management; May/Jun 2008; 22, 3; ABI/INFORM Global pg. 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. RESEARCH Open Access Gastrointestinal symptoms of infantile colic and their change after light needling of acupuncture: a case series study of 913 infants Marianne Reinthal 1,2 , Iréne Lund 3* , Dacil Ullman 4 and Thomas Lundeberg 5 Abstract Background: Infantile colic is a common painful clinical condition associated with signs of distended intestines and an increase in colon peristalsis. However, clinical documentation of observed gastrointestinal functions in the condition is still lacking. Even though the ailment is common, no clear treatment guidelines exist. While acupuncture with minimal stimulation has been shown to be effective in reducing crying behaviour of infants suffering from colic, the documented effect of acupuncture on gastrointestinal function in children with infantile colic is scarce. This case series study aims to document the symptoms of routinely rated gastrointestinal function and the changes in these symptoms after minimal acupuncture in a larger group of children with infantile colic. Methods: This study included 913 infants with normal weights, and lengths at birth. The infants’ mean age was 5.4 weeks when the observations started, and had colic symptoms since two weeks after birth. Light needling stimulation of the acupuncture point LI4 was performed for 10-20 seconds bilaterally on a daily basis for a mean of 6.2 consecutive days. A questionnaire with verbal rating scales for the parents’ evaluation was used before and after the treatment period. Results: Before treatment the infants were assessed by the parent s in terms of ‘often have inflated stomachs’ (99%) and ‘seldom drool’ (76%), ‘regurgitate’ (53%) and ‘belch ’ (62%). Moreover, the reported frequency of defecation was 5-8 times per day (64%), with a yellowish-green colour (61%) and with a water-thin consistency (74%). After treatment, the variables of inflated stomachs, drooling and regurgitating were systematically changed, and rated by the parents as occurring ‘sometimes’ while belching was rated as occurring ‘often’ and the frequency of defecation was reduced to 1-4 times/day with a mustard yellow colour and a gruel-like consistency. The parents also rated their impression of the infants’ general colic symptoms including crying behaviour as much ameliorated in 76% of the cases. Conclusion: The results of the present study show that minimal acupuncture at LI4 in infantile colic is an effective and easy treatment procedure that, furthermore, is reported to be without serious side effects. Background Infantile colic is reported to have an incidence rate between 5% and 19% in prospective studies on infants aged less than three months [1,2]. The classical defini- tion of infantile colic is ‘a seriously fussy or colicky infant who is otherwise healthy and well fed but has paroxysms of irritability and fussing or crying, more than three hours per day, more than three days per week for more than three weeks, or symptoms so severe that medication is indicated’ [3], and this definition is still valid for diagnosis [1]. Thus, the clinical diagnosis is based on the children’ s crying behaviour characterized by paroxysmal and inconsolable crying predominantly in the early night hours and a body language with flexed knees, clenched fists and a grimacing face, often flushed, together expressing a painful state, despite the fact that crying is an unspecified and multifactorial communica- tion of infants, reflecting different reasons for their dis- satisfaction including pain [4]. * Correspondence: Irene.Lund@ki.se 3 Department of Physiology and Pharmacology, Karolinska Institutet, SE-171 77 Stockholm, Sweden Full list of author information is available at the end of the article Reinthal et al. Chinese Medicine 2011, 6:28 http://www.cmjournal.org/content/6/1/28 © 2011 Reinthal et al; licen see BioMed Central Ltd. This is an Open Access article distributed under the terms of the Crea tive Commons Attribution License (http://creativecom ... Arithmetic Series n(a1 + an) Find the sum of each arithmetic series + + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32 4/25 Series and Their Notations 20 + 15 + 10 +…+ −50 12 ∑ 3k − k=1 We are given a1 = and. .. 6/25 Series and Their Notations 1 This problem can be modeled by an arithmetic series with a1 = and d = We are looking for the total number of miles walked after weeks, so we know that n = 8, and. .. infinite series is defined The given formula is not exponential; the series is not geometric because the terms are increasing, and so cannot yield a finite sum 11/25 Series and Their Notations

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