Maths Note

In: Science

Submitted By teambolt1991
Words 573
Pages 3
Series Convergence/Divergence Flow Chart
TEST FOR DIVERGENCE Does limn→∞ an = 0? YES p-SERIES Does an = 1/np , n ≥ 1? NO GEOMETRIC SERIES Does an = arn−1 , n ≥ 1? NO ALTERNATING SERIES Does an = (−1)n bn or an = (−1)n−1 bn , bn ≥ 0? NO TELESCOPING SERIES Do subsequent terms cancel out previous terms in the sum? May have to use partial fractions, properties of logarithms, etc. to put into appropriate form. NO TAYLOR SERIES Does an = f (n)

NO

an Diverges an Converges an Diverges
∞ n=1

YES

Is p > 1?

YES NO

YES

Is |r| < 1?

YES NO

an =

a 1−r

an Diverges

YES

Is bn+1 ≤ bn & lim bn = 0? n→∞ YES

an Converges

YES

Does lim sn = s n→∞ s finite?

YES NO

an = s an Diverges

(a) n! (x

− a) ?

n

YES

Is x in interval of convergence?

YES NO

∞ n=0

an = f (x)

an Diverges

NO Try one or more of the following tests: Is 0 ≤ an ≤ bn ? NO NO Is 0 ≤ bn ≤ an ?


COMPARISON TEST Pick {bn }. Does bn converge?

YES

YES

an Converges

YES

an Diverges an Converges an Diverges
∞ n=a

LIMIT COMPARISON TEST n Pick {bn }. Does lim an = c > 0 n→∞ b c finite & an , bn > 0?

YES

Does n=1 bn converge?

YES NO

INTEGRAL TEST Does an = f (n), f (x) is continuous, positive & decreasing on [a, ∞)? RATIO TEST Is limn→∞ |an+1 /an | = 1?



YES

Does a f (x)dx converge?

YES NO

an Converges

an Diverges an Abs. Conv. an Diverges an Abs. Conv. an Diverges

YES

Is

lim an+1 n→∞ an

< 1?

YES NO

ROOT TEST Is limn→∞ n |an | = 1?

YES

Is lim

n

n→∞

|an | < 1?

YES NO

Problems 1-38 from Stewart’s Calculus, page 784

1.

n2 − 1 n2 + n n=1 n−1 n2 + n n=1 1 2+n n n=1
∞ ∞ ∞







14. n=1 ∞

sin(n) n! 2 · 5 · 8 · · · · · (3n + 2) n=0 n2 + 1 n3 + 1 n=1
∞ ∞

27. k=1 ∞

k ln(k) (k + 1)3

2.

15.

28.

e1/n n2 n=1 tan−1 (n) √ n…...

Similar Documents

Math

...MATH 3330 INFORMATION SHEET FOR FINAL EXAM FALL 2011 FINAL EXAM will be in PKH 103 at 2:00-4:30 pm on Tues Dec 13 • See above for date, time and location of FINAL EXAM. Recall from the first-day handout that any student not obtaining a positive score on the FINAL EXAM will not pass this class. • The material covered will be the same as that covered on the homework from the start of the semester through Dec 6 (but not §6.3) inclusive. (Homework is listed at my website: www.uta.edu/math/vancliff/T/F11 .) • My remaining office hours are: 3:30-4:20 pm on Thurs Dec 8 and 3:30-5:30 pm on Mon Dec 12. • This test will be, in part, multiple choice, but you do NOT need to bring a scantron form. There will be several choices of answer per multiple-choice question and, for each, only one answer will be the correct one. You should do rough work on the test or on paper provided by me. No calculator is allowed. No notes or cards are allowed. BRING YOUR MYMAV ID CARD WITH YOU. • When I write a test, I look over the lecture notes and homework which have already been assigned, and use them to model about 85% of the test problems (and most of them are fair game). You should expect between 30 and 40 questions in total. • A good way to review is to go over the homework problems you have not already done & make sure you understand all the homework well by 48 hours prior to the test. You should also look over the past tests/midterms and understand those fully. In addition,......

Words: 4172 - Pages: 17

Math

...student website. ® Complete the MyMathLab Orientation exercise. ® ® 4/15/13 4/15/13 4/15/13 2 2 Watch this week’s videos located on your student website. Follow the Math Help Community in PhoenixConnect. The focus of the community is to help students succeed in their math courses. Post questions and receive answers from other students, faculty, and staff from the Center for Mathematics Excellence. Resource: Learning Team Toolkit Complete the Learning Team Charter. Complete the Week One assignment in MyMathLab . ® 4/15/13 6 Review your Study Plan in MyMathLab after completing the homework assignment for the week. Select each topic from Ch. 5 in your study plan that has been highlighted with a pushpin • for further review. ® 4/15/13 1 First, complete some Practice problems until you feel ready for a Syllabus 3 MTH/209 Version 6 quiz. o Click the green Practice button within Objectives to Practice and Master. o Complete Practice problems until you feel ready for a quiz. o Click the Close button and return to the main page for the Study Plan. • When you are ready to prove mastery of a concept, click Quiz Me. o Click the Quiz Me button within Objectives to Practice and Master. o Correctly answer 3 of the 4 questions to earn the Mastery Point (MP). Note: If you do not correctly answer 3 of the 4 questions in the Quiz Me, revisit the Practice problems for additional practice, and then retake the Quiz Me until you earn the Mastery......

Words: 3783 - Pages: 16

Math

...MATH 55 SOLUTION SET—SOLUTION SET #5 Note. Any typos or errors in this solution set should be reported to the GSI at isammis@math.berkeley.edu 4.1.8. How many different three-letter initials with none of the letters repeated can people have. Solution. One has 26 choices for the first initial, 25 for the second, and 24 for the third, for a total of (26)(25)(24) possible initials. 4.1.18. How many positive integers less than 1000 (a) are divisible by 7? (b) are divisible by 7 but not by 11? (c) are divisible by both 7 and 11? (d) are divisible by either 7 or 11? (e) are divisible by exactly one of 7 or 11? (f ) are divisible by neither 7 nor 11? (g) have distinct digits? (h) have distinct digits and are even? Solution. (a) Every 7th number is divisible by 7. Since 1000 = (7)(142) + 6, there are 142 multiples of seven less than 1000. (b) Every 77th number is divisible by 77. Since 1000 = (77)(12) + 76, there are 12 multiples of 77 less than 1000. We don’t want to count these, so there are 142 − 12 = 130 multiples of 7 but not 11 less than 1000. (c) We just figured this out to get (b)—there are 12. (d) Since 1000 = (11)(90) + 10, there are 90 multiples of 11 less than 1000. Now, if we add the 142 multiples of 7 to this, we get 232, but in doing this we’ve counted each multiple of 77 twice. We can correct for this by subtracting off the 12 items that we’ve counted twice. Thus, there are 232-12=220 positive integers less than 1000 divisible by 7 or 11. (e) If we want to exclude the......

Words: 3772 - Pages: 16

Math

...STAT2011 Statistical Models sydney.edu.au/science/maths/stat2011 Semester 1, 2014 Computer Exercise Weeks 1 Due by the end of your week 2 session Last compiled: March 11, 2014 Username: mac 1. Below appears the code to generate a single sample of size 4000 from the population {1, 2, 3, 4, 5, 6}. form it into a 1000-by-4 matrix and then find the minimum of each row: > rolls1 table(rolls1) rolls1 1 2 3 4 5 6 703 625 679 662 672 659 2. Next we form this 4000-long vector into a 1000-by-4 matrix: > four.rolls=matrix(rolls1,ncol=4,nrow=1000) 3. Next we find the minimum of each row: > min.roll=apply(four.rolls,1,min) 4. Finally we count how many times the minimum of the 4 rolls was a 1: > sum(min.roll==1) [1] 549 5. (a) First simulate 48,000 rolls: > rolls2=sample(x=c(1,2,3,4,5,6),size=48000,replace=TRUE) > table(rolls2) rolls2 1 2 3 4 5 6 8166 8027 8068 7868 7912 7959 (b) Next we form this into a 2-column matrix (thus with 24,000 rows): > two.rolls=matrix(rolls2,nrow=24000,ncol=2) (c) Here we compute the sum of each (2-roll) row: > sum.rolls=apply(two.rolls,1,sum) > table(sum.rolls) sum.rolls 2 3 4 5 6 7 8 9 10 11 742 1339 2006 2570 3409 4013 3423 2651 1913 1291 1 12 643 Note table() gives us the frequency table for the 24,000 row sums. (d) Next we form the vector of sums into a 24-row matrix (thus with 1,000 columns): > twodozen=matrix(sum.rolls,nrow=24,ncol=1000,byrow=TRUE) (e) To find the 1,000 column minima use > min.pair=apply(twodozen,2,min) (f) Finally compute......

Words: 597 - Pages: 3

Math

...Unit 6: Instructor Graded Assignment Compound Interest In this and future Instructor Graded Assignments you will be asked to use the answers you found in the Unit 1 Assignment. Note: For these questions you need to cite a reliable source for information, which means you cannot use sites like Wikipedia, Ask.com®, and Yahoo® answers. If you do use those sites the instructor may award 0 points for your response. The Assignment problems must have the work shown at all times. The steps for solving the problems must be explained. Failure to do so could result in your submission being given a 0. If you have any questions about how much work to show, please contact your instructor. Assignments must be submitted as a Microsoft Word® document and uploaded to the Dropbox for Unit 6. Please type all answers directly in this Assignment below the question it applies to. All Assignments are due by Tuesday at 11:59 PM ET of the assigned Unit. Note: All interest rates are to be assumed to be yearly interest rates. Question 1 (10 points) 1. You decide to invest $15000 into a bank account that that is compounding its interest monthly. Assuming the bank is paying out an interest rate of the current prime rate - 1% (In the event that prime - 1% is less than 1%, use 1%), and the investment is for 5 years a) How much money (total) do you have after the 5 years pass? Total = A(1+r/n)nt A = $15,000, r = 0.01 , n = 12, t = 5 Total = $......

Words: 886 - Pages: 4

Math

...This article is about the study of topics, such as quantity and structure. For other uses, see Mathematics (disambiguation). "Math" redirects here. For other uses, see Math (disambiguation). Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.[1] Mathematics is the study of topics such as quantity (numbers),[2] structure,[3] space,[2] and change.[4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.[7][8] Mathematicians seek out patterns[9][10] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become......

Words: 634 - Pages: 3

Math

...method in general. This will be followed by a more detailed description of how the scientific method will be applied in this course. In the end, this discussion will give you the format for the lab reports you will write over the next 15 weeks. Our first lab will focus on taking good measurements, graphing techniques, and how to extract the important information from the data to reach a conclusion. You will be measuring round objects, so you are invited to bring different-sized round objects to class on the first day. In preparation for class, consider how round objects can be measured. 1) Brainstorm to develop the testable question. How does one measure the roundness of a round object? 2) Develop the hypothesis. Note: While the relationship in this lab is very simple and should be well known, this will not always be the case. This lab is meant to introduce the lab format without too many complications. If the diameter is increased, then the circumference is increased in a directly linear proportional manner. 3) Brainstorm for variables: The two variables that make up the testable question must be included. Identify the variables. Independent: Diameter Dependent: Circumference Control: Round objects 4) Develop the design table. | |D(cm) |C(cm) | | | | ......

Words: 730 - Pages: 3

Math

...the schedule below to see the topics that will make up the material for each exam. With a valid written excuse and making immediate arrangements with the instructor, a missed exam might be replaced with the grade of the final exam and/or the average grade of all tests (including final) and/or quizzes. * Attendance: It is the university policy that attendance is compulsory. A student missing more than 15% (7 UTR classes or 5 MW classes) of the total allocated course hours will receive a WF. | N | Student Academic Integrity Code Statement | Student must adhere  to the Academic Integrity code stated in the 2014-2015 undergraduate catalog | SCHEDULE Note: Tests and other graded assignments due dates are set. No addendum, make-up exams, or extra assignments to improve grades will be given. # | WEEK | CHAPTER/SECTIONS | NOTES | 1 | Sept 14 - Sept 18 | 5: Review of Formulas and Integration Techniques7.1: Integration by Parts | | 2 | Sept 21 - Sept 25 | 7.2: Trigonometric Integrals7.3: Trigonometric Substitution | | 3 | Sept 28 – Oct 2 | 7.4: Integration of Rational Functions by Partial Fractions 7.5: Strategy for Integration | | | Oct 5– Oct 9 | Holiday (Eid Al Adha) | | 4 | Oct 12– Oct 16 | 7.8: Improper Integrals | | 5 | Oct 19 – Oct 23 | 8.1: Arc Length8.2: Area of a Surface of Revolution | | 6 | Oct 26 – Oct 30 | 10.1: Parametric Equations10.2: Calculus with Parametric Curves | | 7 | Nov 2 – Nov 6 | Review for Midterm Exam 1Midterm Exam 1 | Exam 1......

Words: 1366 - Pages: 6

Math

...can see how to split up the original equation into its factor pair, this is the quickest and allows you to solve the problem in one step. Week 9 capstone part 1 Has the content in this course allowed you to think of math as a useful tool? If so, how? What concepts investigated in this course can apply to your personal and professional life? In the course, I have learned about polynomials, rational expressions, radical equations, and quadratic equations. Quadratic equations seem to have the most real life applications -- in things such as ticket sales, bike repairs, and modeling. Rational expressions are also important, if I know how long it takes me to clean my sons room, and know how long it takes him to clean his own room. I can use rational expressions to determine how long it will take the two of us working together to clean his room. The Math lab site was useful in some ways, since it allowed me to check my answers to the problems immediately. However, especially in math 117, it was too sensitive to formatting of the equations and answers. I sometimes put an answer into the math lab that I knew was right, but it marked it wrong because of the math lab expecting slightly different formatting Week 9 capstone part 2 I really didn't use center for math excellence because i found that MML was more convenient for me. I think that MML reassures you that you’re doing the problem correctly. MML is extra support because it carefully walks you through the problem visually......

Words: 662 - Pages: 3

Math

...and solve problems in everyday life”. In my everyday life I have to keep the balance in my check book, pay bills, take care of kids, run my house, cook, clean etc. With cooking I am using math, measuring how much food to make for four people (I still haven’t mastered that one). With bills I am using math, how much each company gets, to how much money I have to spare (which these days is not much). In my everyday life I do use some form of a math. It might not be how I was taught, but I have learned to adapt to my surroundings and do math how I know it be used, the basic ways, none of that fancy stuff. For my weakest ability I would say I fall into “Confidence with Mathematics”. Math has never been one of my favorite subjects to learn. It is like my brain knows I have to learn it, but it puts up a wall and doesn’t allow the information to stay in there. The handout “The Case for Quantitative Literacy” states I should be at ease with applying quantitative methods, and comfortable with quantitative ideas. To be honest this class scares the crap out of me, and I am worried I won’t do well in this class. The handout also says confidence is the opposite of “Math Anxiety”, well I can assure you I have plenty of anxiety right now with this class. I have never been a confident person with math, I guess I doubt my abilities, because once I get over my fears and anxiety I do fine. I just have to mentally get myself there and usually it’s towards the end of the class. There are......

Words: 418 - Pages: 2

Notes

...accounting issues, where relevant. You should also evaluate the financial condition and performance of each company. You may wish to include projected financial statements and a valuation based on those projections. However, finance covers valuation in far more detail in finance, so valuation is not the crucial part of this write-up. Each group will have about 45 minutes for their presentation, depending on the number of groups. That will include five minutes for setup, twenty minutes for a presentation, and five minutes for questions. 4. Final Exam. The exam will consist of five or six questions taken from footnotes, financial statements, or articles. The exam may include any topic covered in this course. The exam is open-book, open-notes, and you may use computers, calculators, and the Internet (not including e-mail). Grading: Grading is based on a mid-term and final examination, a group case presentation, a group project, and class participation. Class participation grading begins Monday, 14 August; class participation in the accounting camp does not count toward your final grade. Grade weighting is as follows: Group project. Write-up and presentation 40% Final examination 35% Class participation 25% I assign numerical grades for each assignment (maximum of 100 points for each assignment). At the end of the term I weight scores by the above percentages, add scores for the four assignments, and rank scores from high to low. I......

Words: 2809 - Pages: 12

Math

... | | |Location  | | |on-line | | | | | |Times  | | |on-line, or in Maier Hall , Math Lab, Peninsula College | | | | | |Start Date  | | |Sept. 21, 2015 End Date Dec. 9, 2015 | | | | | |Course Credits  ...

Words: 1789 - Pages: 8

Math Notes

...1. Read Module 3 Topic 7, Module 4 Topic 1 and 2, Module 5 Topic 1~2. 2. Do the drills for the topics. 3. Read the Chapter 3 sections 2, 5 and Chapter 4 sections 1~3 in your textbook. 4. Do Homework for week 5 (you can find the list in the conference). Week 5 Supplementary Notes Chapter 3 Section 3.2: Polynomial Function of Higher Degree A polynomial function P is given by , where the coefficients are real numbers and the exponents are whole numbers. This polynomial is of nth degree. Far-Left and Far-Right Behavior The behavior of the graph of a polynomial function as x becomes very large or very small is referred to as the end behavior of the graph. The leading term of a polynomial function determines its end behavior. x becomes very large x → ∞ x becomes very large x → ∞ x becomes very small -∞ ← x x becomes very small -∞ ← x We can summarize the end behavior as follows: The Leading-Term Test If is the leading term of a polynomial, then the behavior of the graph as x → ∞ or as x → −∞ can be described in one of the four following ways. If n is even and an >0: ▼ ▼ | If n is even and an <0:▲ ▲ | If n is odd and an >0: ▲▼ | If n is odd and an <0: ▲ ▼ | Polynomial Function, Real Zeros, Graphs, and Factors (x − c) If c is a real zero of a function (that is, f(c)=0), then (c,0) is an x-intercept of the......

Words: 1629 - Pages: 7

Strategies for Math and Note Taking

...Bianca Fields Miss Dibble SDS101 9 November 2015 Strategies for Math and Note Taking I believe Math is learned by doing the problems and doing the homework. The problems help you learn the formulas you need to know, to help with problem solving. I have learned from my own personal experience that you must keep up with the Instructor: attend class, read the text and do homework every day. Falling a day behind puts you at a disadvantage. Falling a week behind puts you in deep trouble. I have found that college Math is much different than high school math. College math class meets less often and covers material at about twice the pace than a High School course does. You are expected to understand new material much more quickly. So it is very important that if there is something you don’t understand, you should take responsibility for studying, recognizing what you do and don't know, and knowing how to get your Instructor to help you with what you don't know. When it comes to Math testing, it can be very stressful, as with taking any test. Some of the strategies, I have tried, is to look for the problems I definitely know how to do right away, and those that I have to think about. I start with the problems that I know for sure I can do. Then I try the problems I think I can figure out, then I finally try the problems I’m least sure about. I try to make sure I read the questions carefully, and do all parts of each problem. If I find that I am stuck on a problem, I......

Words: 638 - Pages: 3

Maths for Business and Economics Notes

... amn    then  is a matrix with dim  c11 c12  c1 p  c n c22  c2 p   21  , where c  a b C  ik kj ij      k 1    cm1 cm 2  cmp  • Note, for  to be meaningful, the number of columns of  should be equal to the  number of rows of  6 Examples • • 1 3 2 4 Let  5 7 6 1 = 8 3 1 4 2 5 7 19 5 5 3 ,  0 25 ,   28 2 4 7 7 1 3 6 6 1 3 0 2 4 , then 5 9 12 19 26 20 25 8 19 = 8 43 2 4 22 50 3 9 0 • It is important to note that usually,  • Even if  .   exists,  may not be defined, for example,  1 2 5 ,  3 4 6 7 Useful properties ′ ′ ′ • Transpose of product:   • Left distributive law:   • Right distributive law:   • Associative law:   • Exercise:  Prove the above properties for 2×2 matrices  , and .  ∙ • Exercise:  Confirm the following three points by example – – – ,     except in special cases 0 does not imply that  or  is 0 and  0 do not imply that  8 Trace of a square matrix • Trace of square matrix is the sum of its diagonal elements n tr ( A)   aii i 1 • • For square matrices  and  , tr ( A  B )  tr ( A)  tr ( B ) tr ( A ')  tr ( A) .  (note, both   n  tr ( AB )   ( AB )ii     aik bki  i 1 i 1  k 1  n n m m n   aik bki   ( BA) kk  tr ( BA) k 1 i 1 k......

Words: 6927 - Pages: 28

Android TVボックス&アクセサリー | Electric Meat Grinder | Top wo Nerae! Gunbuster Episode 2 English Subbed