Sequences, Series and Convergence with the TI 92. Roger G. Brown Monash University

Size: px
Start display at page:

Download "Sequences, Series and Convergence with the TI 92. Roger G. Brown Monash University"

Transcription

1 Sequeces, Series ad Covergece with the TI 92. Roger G. Brow Moash Uiversity Itroductio. Studets erollig i calculus at Moash Uiversity, like may other calculus courses, are itroduced to sequeces ad series i the same sectio of work. These studets have difficulty distiguishig betwee a sequece ad a series. These difficulties are compouded by the eed for a sophisticated level of algebra as well as a soud kowledge of limits whe dealig with covergece of a sequece or a series. The difficulties have bee recogised by other authors icludig Morrel (992) The studets completig the first semester of the first year calculus course at Moash Uiversity, agai demostrated a poor uderstadig of the differece betwee a sequece ad a series as well as determiig whether a series was coverget or ot. I the secod semester, a group of studets icludig some who were repeatig the subject, were provided with a ivestigative approach to the topic by utilisig the multiple represetatios of the TI 92. The studets were ot oly able to graph ad tabulate sequeces, partial sums of series, they were also able to compare algebraically derived limits for both sequeces ad series. The ability to use the symbolic algebra properties of the TI 92 further assisted the studets i developig a uderstadig of the differece betwee a sequece ad a series. Studet Activities. A sample studet activity follows: The Geome Tree A demostratio usig Cabri o the TI 92 was used to costruct the sequece of Geome Trees. Studets commeced by coutig the umber of segmets. Most of the studets were able to determie the sequece for the legth of the ew segmets added, as well as write the formula for the th term. The studets were the asked to determie the partial sums for the legth of a path, from the base to the ed of a brach, before writig the series for the legth of a path. Fially they were also required to determie the partial sums ad the series for the umber of segmets.

2 Oce the studets had developed the sequece formula for the legth of the brach ad the series formula for the both the legth of the path ad the umber of braches they were the able to plot the graphs usig the sequece mode of the TI 92. A visual ispectio of the tables ad graphs allowed the studets to see whether the series coverged or diverged. The legths of the ew segmet added show both by the table ad the graph. l r = r 2 The partial sums for the legth of the path where represets the umber of braches are also show both by the table ad the graph. The formula for the partial sum is give by p = r r = 2 The partial sums for the umber of braches is also show, by both the table ad the graph. The formula for the partial sums is give by r b = 2 r = This ituitive itroductio allowed the studets to firstly develop a uderstadig of covergece ad divergece ad prepared them for a ivestigatio ivolvig the use of limits o the TI 92. By usig the split scree mode of the TI 92 they were able to view both the graph of a sequece (icludig a sequece of partial sums) ad a table of values for the terms of a sequece. The compariso of a umber 2

3 of examples eabled them to ituitively determie that the limit of the sequece of terms for a covergig series was 0, while for a diverget series ay value was possible icludig 0 ad. By comparig the graph of the partial sums of the legth of the path with the limit of the sequece it was see by the studets that the series is covergig ad that the limit of the correspodig sequece of terms is zero. The graph of the partial sums of the total umber of segmets portrays the series as diverget ad this is cofirmed by the limit of the correspodig sequece of terms which is ot equal to zero. The use of cotextual examples is importat as it provided studets with visual support to assist the developmet of their uderstadig of the terms sequece, partial sum ad series, as well as experiece with the summatio sig. Havig completed cotextual examples related to the Koch Sowflake ad the Sierpiska triagle they the moved oto examples ivolvig other covergig or divergig series such as the followig Example. Cosider the series. Plot the sequece of partial sums ad the ( + ) = determie whether the series coverges or ot. 3

4 Example 2. Cosider the series. Plot the sequece of partial sums ad the + = determie whether the series coverges or ot. I this example it was importat for the studets to recogise that the sequece cosistig of the terms of the series coverges to ad therefore the series is diverget. RESULTS At the coclusio of the sessios o Sequeces ad Series the studets were give a test coverig the work completed. A portio of the questios were the same as o the test i first semester, while others were adapted to suit the differet approach used i secod semester. Uiversity policy determied that the studets were oly allowed to use scietific calculators i the test. The results for the comparative questios are show i Table. 4

5 Questio. lim Determie whether x 0, exists or x 2 ot. Calculate the limit if it exists. Questio 2. Determie whether the followig sequece coverges or ot. If it coverges fid the value that it coverges to. = Questio 3. State whether the followig series diverges or coverges, givig reasos = First semester % correct = 43 23% ot applicable 23% Secod semester % correct = 28 8% 7% 54% Table. Comparative results from semester ad 2 for similar questios. Whe comparig the performaces from questios 2 ad 3 i semester two, oly studets or 39 % of the group were able to provide correct solutios to both questios, idicatig that studets were still havig difficulty whe comparig the covergece of a sequece ad that of a series. The results i the table demostrate that over half of those studets tested were able to recogise ad justify the correct solutio for questio 3 which is a marked icrease o the first semesters result. Studets were also required to costruct cocept maps usig at least the followig terms: sequece, series, partial sums, limits, covergece ad divergece. For may of the studets this was the first time they had bee asked to draw a cocept map. These cocept maps were scored accordig to the methods of Novak ad Gowi (984) ad the the scores from the cocept maps were compared with the studets resposes o the test. The studets' performace o the test was reflected i their costructio of the cocept maps. Studets lack of uderstadig of the differece betwee the covergece of a sequece ad that of a series was evidet. By cosiderig the cocept maps ad the test results it appeared that the studets' difficulty lay i the otio of fidig the limit as approached ifiity for the covergece of a sequece, i cotrast with the limit as approached ifiity for the terms of a series beig zero whe a series coverges. 5

6 Coclusio The effects of the work completed usig both a cotextual base ad the Texas TI 92 has ehaced the performace of studets i the first year calculus course at Moash Uiversity. Although the results from this alterative method of istructio are ecouragig, further research will eed to be udertake to determie the ogoig effectiveess of these methods. Particularly by developig the studets ability to recogise the uiqueess of the limit whe cosiderig the covergece or divergece of a series. Refereces Morrel, J. H. (992). A getle itroductio to ifiite series usig a graphig calculator. Primus 992 Vol 2 (). Novak, J.D. & Gowi, D. B. (984). Learig how to lear. Cambridge Uiversity Press: Cambridge. 6

Lesson 12. Sequences and Series

Lesson 12. Sequences and Series Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or

More information

Section 9.2 Series and Convergence

Section 9.2 Series and Convergence Sectio 9. Series ad Covergece Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

4.3. The Integral and Comparison Tests

4.3. The Integral and Comparison Tests 4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece

More information

3.2 Introduction to Infinite Series

3.2 Introduction to Infinite Series 3.2 Itroductio to Ifiite Series May of our ifiite sequeces, for the remaider of the course, will be defied by sums. For example, the sequece S m := 2. () is defied by a sum. Its terms (partial sums) are

More information

TILE PATTERNS & GRAPHING

TILE PATTERNS & GRAPHING TILE PATTERNS & GRAPHING LESSON 1 THE BIG IDEA Tile patters provide a meaigful cotext i which to geerate equivalet algebraic expressios ad develop uderstadig of the cocept of a variable. Such patters are

More information

2.3. GEOMETRIC SERIES

2.3. GEOMETRIC SERIES 6 CHAPTER INFINITE SERIES GEOMETRIC SERIES Oe of the most importat types of ifiite series are geometric series A geometric series is simply the sum of a geometric sequece, Fortuately, geometric series

More information

G r a d e. 5 M a t h e M a t i c s. Patterns and relations

G r a d e. 5 M a t h e M a t i c s. Patterns and relations G r a d e 5 M a t h e M a t i c s Patters ad relatios Grade 5: Patters ad Relatios (Patters) (5.PR.1) Edurig Uderstadigs: Number patters ad relatioships ca be represeted usig variables. Geeral Outcome:

More information

Basic Elements of Arithmetic Sequences and Series

Basic Elements of Arithmetic Sequences and Series MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

More information

The geometric series and the ratio test

The geometric series and the ratio test The geometric series ad the ratio test Today we are goig to develop aother test for covergece based o the iterplay betwee the it compariso test we developed last time ad the geometric series. A ote about

More information

Unit 2 Sequences and Series

Unit 2 Sequences and Series Accelerated Mathematics III Frameworks Studet Editio Uit Sequeces ad Series d Editio April, 011 Table of Cotets INTRODUCTION:... 3 Reaissace Festival Learig Task... 8 Fasciatig Fractals Learig Task...

More information

Unit 2 Sequences and Series

Unit 2 Sequences and Series Mathematics IV Uit 1 st Editio Mathematics IV Frameworks Studet Editio Uit Sequeces ad Series 1 st Editio Kathy Cox, State Superitedet of Schools Uit : Page 1 of 35 Mathematics IV Uit 1 st Editio Table

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

Geometric Sequences and Series. Geometric Sequences. Definition of Geometric Sequence. such that. a2 4

Geometric Sequences and Series. Geometric Sequences. Definition of Geometric Sequence. such that. a2 4 3330_0903qxd /5/05 :3 AM Page 663 Sectio 93 93 Geometric Sequeces ad Series 663 Geometric Sequeces ad Series What you should lear Recogize, write, ad fid the th terms of geometric sequeces Fid th partial

More information

The Harmonic Series Diverges Again and Again

The Harmonic Series Diverges Again and Again The Harmoic Series Diverges Agai ad Agai Steve J. Kifowit Prairie State College Terra A. Stamps Prairie State College The harmoic series, = = 3 4 5, is oe of the most celebrated ifiite series of mathematics.

More information

Theorems About Power Series

Theorems About Power Series Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Mocks.ie Maths LC HL Further Calculus mocks.ie Page 1

Mocks.ie Maths LC HL Further Calculus mocks.ie Page 1 Maths Leavig Cert Higher Level Further Calculus Questio Paper By Cillia Fahy ad Darro Higgis Mocks.ie Maths LC HL Further Calculus mocks.ie Page Further Calculus ad Series, Paper II Q8 Table of Cotets:.

More information

Arithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,...

Arithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,... 3330_090.qxd 1/5/05 11:9 AM Page 653 Sectio 9. Arithmetic Sequeces ad Partial Sums 653 9. Arithmetic Sequeces ad Partial Sums What you should lear Recogize,write, ad fid the th terms of arithmetic sequeces.

More information

MA2108S Tutorial 5 Solution

MA2108S Tutorial 5 Solution MA08S Tutorial 5 Solutio Prepared by: LuJigyi LuoYusheg March 0 Sectio 3. Questio 7. Let x := / l( + ) for N. (a). Use the difiitio of limit to show that lim(x ) = 0. Proof. Give ay ɛ > 0, sice ɛ > 0,

More information

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

1. a n = 2. a n = 3. a n = 4. a n = 5. a n = 6. a n =

1. a n = 2. a n = 3. a n = 4. a n = 5. a n = 6. a n = Versio PREVIEW Homework Berg (5860 This prit-out should have 9 questios. Multiple-choice questios may cotiue o the ext colum or page fid all choices before aswerig. CalCb0b 00 0.0 poits Rewrite the fiite

More information

Building Blocks Problem Related to Harmonic Series

Building Blocks Problem Related to Harmonic Series TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite

More information

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

More information

8.5 Alternating infinite series

8.5 Alternating infinite series 65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,

More information

Department of Computer Science, University of Otago

Department of Computer Science, University of Otago Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

More information

5. SEQUENCES AND SERIES

5. SEQUENCES AND SERIES 5. SEQUENCES AND SERIES 5.. Limits of Sequeces Let N = {0,,,... } be the set of atural umbers ad let R be the set of real umbers. A ifiite real sequece u 0, u, u, is a fuctio from N to R, where we write

More information

The Limit of a Sequence

The Limit of a Sequence 3 The Limit of a Sequece 3. Defiitio of limit. I Chapter we discussed the limit of sequeces that were mootoe; this restrictio allowed some short-cuts ad gave a quick itroductio to the cocept. But may importat

More information

G r a d e. 2 M a t h e M a t i c s. statistics and Probability

G r a d e. 2 M a t h e M a t i c s. statistics and Probability G r a d e 2 M a t h e M a t i c s statistics ad Probability Grade 2: Statistics (Data Aalysis) (2.SP.1, 2.SP.2) edurig uderstadigs: data ca be collected ad orgaized i a variety of ways. data ca be used

More information

ORDERS OF GROWTH KEITH CONRAD

ORDERS OF GROWTH KEITH CONRAD ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus

More information

MESSAGE TO TEACHERS: NOTE TO EDUCATORS:

MESSAGE TO TEACHERS: NOTE TO EDUCATORS: MESSAGE TO TEACHERS: NOTE TO EDUCATORS: Attached herewith, please fid suggested lesso plas for term 1 of MATHEMATICS Grade 12. Please ote that these lesso plas are to be used oly as a guide ad teachers

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

Modified Line Search Method for Global Optimization

Modified Line Search Method for Global Optimization Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

More information

Grade 7. Strand: Number Specific Learning Outcomes It is expected that students will:

Grade 7. Strand: Number Specific Learning Outcomes It is expected that students will: Strad: Number Specific Learig Outcomes It is expected that studets will: 7.N.1. Determie ad explai why a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad why a umber caot be divided by 0. [C, R] [C]

More information

Strategy for Testing Series

Strategy for Testing Series Strategy for Testig Series We ow have several ways of testig a series for covergece or divergece; the problem is to decide which test to use o which series. I this respect testig series is similar to itegratig

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios

More information

8.4. Click here for solutions. Click here for answers. OTHER CONVERGENCE TESTS. 3 n. 2n 1! sn 3. 2 n n 2. 3n n 1. 1 n 1 5 n 1 n n 2

8.4. Click here for solutions. Click here for answers. OTHER CONVERGENCE TESTS. 3 n. 2n 1! sn 3. 2 n n 2. 3n n 1. 1 n 1 5 n 1 n n 2 SECTION OTHER CONVERGENCE TESTS OTHER CONVERGENCE TESTS A Click here for aswers. S Click here for solutios. 4 Test the series for covergece or divergece.. 2. 3. 2 2 3 3 4 4 5 5 6 6 7 4. 5. 6. 7. 5 8. 9.

More information

Bridging Units: Resource Pocket 4

Bridging Units: Resource Pocket 4 Bridgig Uits: Resource Pocket 4 Iterative methods for solvig equatios umerically This pocket itroduces the cocepts of usig iterative methods to solve equatios umerically i cases where a algebraic approach

More information

Math 115 HW #4 Solutions

Math 115 HW #4 Solutions Math 5 HW #4 Solutios From 2.5 8. Does the series coverge or diverge? ( ) 3 + 2 = Aswer: This is a alteratig series, so we eed to check that the terms satisfy the hypotheses of the Alteratig Series Test.

More information

STUDENTS PARTICIPATION IN ONLINE LEARNING IN BUSINESS COURSES AT UNIVERSITAS TERBUKA, INDONESIA. Maya Maria, Universitas Terbuka, Indonesia

STUDENTS PARTICIPATION IN ONLINE LEARNING IN BUSINESS COURSES AT UNIVERSITAS TERBUKA, INDONESIA. Maya Maria, Universitas Terbuka, Indonesia STUDENTS PARTICIPATION IN ONLINE LEARNING IN BUSINESS COURSES AT UNIVERSITAS TERBUKA, INDONESIA Maya Maria, Uiversitas Terbuka, Idoesia Co-author: Amiuddi Zuhairi, Uiversitas Terbuka, Idoesia Kuria Edah

More information

G r a d e. 5 M a t h e M a t i c s. shape and space

G r a d e. 5 M a t h e M a t i c s. shape and space G r a d e 5 M a t h e M a t i c s shape ad space Grade 5: Shape ad Space (Measuremet) (5.SS.1) Edurig Uderstadigs: there is o direct relatioship betwee perimeter ad area. Geeral Outcome: Use direct or

More information

1 The Binomial Theorem: Another Approach

1 The Binomial Theorem: Another Approach The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets

More information

CALCULUS. Taimur Khalid

CALCULUS. Taimur Khalid CALCULUS Taimur Khalid 5/20/2015 Chapter 1 What is Calculus? What is it? I m sure you ve all already heard about what calculus is from the other studets i my calculus class, but just i case you eed a refresher,

More information

Binet Formulas for Recursive Integer Sequences

Binet Formulas for Recursive Integer Sequences Biet Formulas for Recursive Iteger Sequeces Homer W. Austi Jatha W. Austi Abstract May iteger sequeces are recursive sequeces ad ca be defied either recursively or explicitly by use of Biet-type formulas.

More information

Numerical Solution of Equations

Numerical Solution of Equations School of Mechaical Aerospace ad Civil Egieerig Numerical Solutio of Equatios T J Craft George Begg Buildig, C4 TPFE MSc CFD- Readig: J Ferziger, M Peric, Computatioal Methods for Fluid Dyamics HK Versteeg,

More information

Review for College Algebra Final Exam

Review for College Algebra Final Exam Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 1-4. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i

More information

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or p-series (the Compariso Test), but of

More information

Approximating the Sum of a Convergent Series

Approximating the Sum of a Convergent Series Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA 30030 lriddle@agesscott.edu The BC Calculus Course Descriptio metios how techology ca be used to explore covergece

More information

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers . Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,

More information

represented by 4! different arrangements of boxes, divide by 4! to get ways

represented by 4! different arrangements of boxes, divide by 4! to get ways Problem Set #6 solutios A juggler colors idetical jugglig balls red, white, ad blue (a I how may ways ca this be doe if each color is used at least oce? Let us preemptively color oe ball i each color,

More information

B1. Fourier Analysis of Discrete Time Signals

B1. Fourier Analysis of Discrete Time Signals B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT)

More information

G r a d e 6 M a t h e m a t i c s. Shape and Space

G r a d e 6 M a t h e m a t i c s. Shape and Space G r a d e 6 M a t h e m a t i c s Shape ad Space Grade 6: Shape ad Space (Measuremet) (6.SS.1, 6.SS.2) Edurig Uderstadig(s): All measuremets are comparisos. The uit of measure must be of the same ature

More information

Section IV.5: Recurrence Relations from Algorithms

Section IV.5: Recurrence Relations from Algorithms Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by

More information

Multiple Representations for Pattern Exploration with the Graphing Calculator and Manipulatives

Multiple Representations for Pattern Exploration with the Graphing Calculator and Manipulatives Douglas A. Lapp Multiple Represetatios for Patter Exploratio with the Graphig Calculator ad Maipulatives To teach mathematics as a coected system of cocepts, we must have a shift i emphasis from a curriculum

More information

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is 0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

More information

The second difference is the sequence of differences of the first difference sequence, 2

The second difference is the sequence of differences of the first difference sequence, 2 Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

More information

Learning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.

Learning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr. Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the

More information

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51 Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log

More information

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients 652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

More information

INFINITE SERIES KEITH CONRAD

INFINITE SERIES KEITH CONRAD INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal

More information

7.1 Finding Rational Solutions of Polynomial Equations

7.1 Finding Rational Solutions of Polynomial Equations 4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?

More information

Your grandmother and her financial counselor

Your grandmother and her financial counselor Sectio 10. Arithmetic Sequeces 963 Objectives Sectio 10. Fid the commo differece for a arithmetic sequece. Write s of a arithmetic sequece. Use the formula for the geeral of a arithmetic sequece. Use the

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say

More information

Arithmetic Sequences

Arithmetic Sequences . Arithmetic Sequeces Essetial Questio How ca you use a arithmetic sequece to describe a patter? A arithmetic sequece is a ordered list of umbers i which the differece betwee each pair of cosecutive terms,

More information

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread and Boxplots Discrete Math, Section 9.4 Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

More information

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

More information

SEQUENCES AND SERIES. Chapter Nine

SEQUENCES AND SERIES. Chapter Nine Chapter Nie SEQUENCES AND SERIES I this chapter, we look at ifiite lists of umbers, called sequeces, ad ifiite sums, called series. I Sectio 9., we study sequeces. I Sectio 9.2, we begi with a particular

More information

23.3 Sampling Distributions

23.3 Sampling Distributions COMMON CORE Locker LESSON Commo Core Math Stadards The studet is expected to: COMMON CORE S-IC.B.4 Use data from a sample survey to estimate a populatio mea or proportio; develop a margi of error through

More information

Estimating the Mean and Variance of a Normal Distribution

Estimating the Mean and Variance of a Normal Distribution Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers

More information

ARITHMETIC AND GEOMETRIC PROGRESSIONS

ARITHMETIC AND GEOMETRIC PROGRESSIONS Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives

More information

Chapter Eleven. Taylor Series. (x a) k. c k. k= 0

Chapter Eleven. Taylor Series. (x a) k. c k. k= 0 Chapter Eleve Taylor Series 111 Power Series Now that we are kowledgeable about series, we ca retur to the problem of ivestigatig the approximatio of fuctios by Taylor polyomials of higher ad higher degree

More information

Intro to Sequences / Arithmetic Sequences and Series Levels

Intro to Sequences / Arithmetic Sequences and Series Levels Itro to Sequeces / Arithmetic Sequeces ad Series Levels Level : pg. 569: #7, 0, 33 Pg. 575: #, 7, 8 Pg. 584: #8, 9, 34, 36 Levels, 3, ad 4(Fiboacci Sequece Extesio) See Hadout Check for Uderstadig Level

More information

Math : Sequences and Series

Math : Sequences and Series EP-Program - Strisuksa School - Roi-et Math : Sequeces ad Series Dr.Wattaa Toutip - Departmet of Mathematics Kho Kae Uiversity 00 :Wattaa Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou. Sequeces

More information

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means) CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

More information

MATH 140A - HW 5 SOLUTIONS

MATH 140A - HW 5 SOLUTIONS MATH 40A - HW 5 SOLUTIONS Problem WR Ch 3 #8. If a coverges, ad if {b } is mootoic ad bouded, rove that a b coverges. Solutio. Theorem 3.4 states that if a the artial sums of a form a bouded sequece; b

More information

Module 4: Mathematical Induction

Module 4: Mathematical Induction Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

More information

Chapter 5: Inner Product Spaces

Chapter 5: Inner Product Spaces Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

More information

AP Calculus BC 2003 Scoring Guidelines Form B

AP Calculus BC 2003 Scoring Guidelines Form B AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet

More information

Sequences II. Chapter 3. 3.1 Convergent Sequences

Sequences II. Chapter 3. 3.1 Convergent Sequences Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

More information

8.3 POLAR FORM AND DEMOIVRE S THEOREM

8.3 POLAR FORM AND DEMOIVRE S THEOREM SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 48 8. POLAR FORM AND DEMOIVRE S THEOREM Figure 8.6 (a, b) b r a 0 θ Complex Number: a + bi Rectagular Form: (a, b) Polar Form: (r, θ) At this poit you ca add,

More information

Students Knowledge and Strategies for Solving Equations

Students Knowledge and Strategies for Solving Equations Studets Kowledge ad Strategies for Solvig Equatios Chris Lisell Uiversity of Otago College of Educatio This paper presets results from the secod year of a ivestigatio ito studets

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

Lesson 17 Pearson s Correlation Coefficient

Lesson 17 Pearson s Correlation Coefficient Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

More information

Lecture 7: Borel Sets and Lebesgue Measure

Lecture 7: Borel Sets and Lebesgue Measure EE50: Probability Foudatios for Electrical Egieers July-November 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,

More information

Riemann Sums y = f (x)

Riemann Sums y = f (x) Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid

More information

Tests for Convergence of Series. a n > 1 n. 0 < a n < 1 n 2. 0 < a n <.

Tests for Convergence of Series. a n > 1 n. 0 < a n < 1 n 2. 0 < a n <. Tests for Covergece of Series ) Use the compari test to cofirm the statemets i the followig eercises.. 4 diverges, 4 3 diverges. Aswer: Let a / 3), for 4. Sice 3 /, a >. The harmoic series

More information

Understanding Rational Exponents and Radicals

Understanding Rational Exponents and Radicals x Locker LESSON. Uderstadig Ratioal Expoets ad Radicals Name Class Date. Uderstadig Ratioal Expoets ad Radicals Essetial Questio: How are radicals ad ratioal expoets related? A..A simplify umerical radical

More information

Solving Inequalities

Solving Inequalities Solvig Iequalities Say Thaks to the Authors Click http://www.ck12.org/saythaks (No sig i required) To access a customizable versio of this book, as well as other iteractive cotet, visit www.ck12.org CK-12

More information

Standard Errors and Confidence Intervals

Standard Errors and Confidence Intervals Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5-year-old boys. If we assume

More information

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics

More information

Linear Algebra II. Notes 6 25th November 2010

Linear Algebra II. Notes 6 25th November 2010 MTH6140 Liear Algebra II Notes 6 25th November 2010 6 Quadratic forms A lot of applicatios of mathematics ivolve dealig with quadratic forms: you meet them i statistics (aalysis of variace) ad mechaics

More information

CHUTES AND LADDERS MICHAEL HOCHMAN

CHUTES AND LADDERS MICHAEL HOCHMAN CHUTES AND LADDERS MICHAEL HOCHMAN Abstract. This paper discusses ad the uses the theory of Markov chais to aalyze ad develop a theory of the board game Chutes ad Ladders. Further, a method of usig computer

More information

Essential Question How can you use properties of exponents to simplify products and quotients of radicals?

Essential Question How can you use properties of exponents to simplify products and quotients of radicals? . Properties of Ratioal Expoets ad Radicals Essetial Questio How ca you use properties of expoets to simplify products ad quotiets of radicals? Reviewig Properties of Expoets Work with a parter. Let a

More information

Contents. 7 Sequences and Series. 7.1 Sequences and Convergence. Calculus II (part 3): Sequences and Series (by Evan Dummit, 2015, v. 2.

Contents. 7 Sequences and Series. 7.1 Sequences and Convergence. Calculus II (part 3): Sequences and Series (by Evan Dummit, 2015, v. 2. Calculus II (part 3): Sequeces ad Series (by Eva Dummit, 05, v..00) Cotets 7 Sequeces ad Series 7. Sequeces ad Covergece......................................... 7. Iite Series.................................................

More information

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

More information

Descriptive statistics deals with the description or simple analysis of population or sample data.

Descriptive statistics deals with the description or simple analysis of population or sample data. Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small

More information

Lesson 15 ANOVA (analysis of variance)

Lesson 15 ANOVA (analysis of variance) Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi

More information

9.8: THE POWER OF A TEST

9.8: THE POWER OF A TEST 9.8: The Power of a Test CD9-1 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based

More information

Infinite Sequences and Series

Infinite Sequences and Series CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

More information