Approximating the Sum of a Convergent Series

Size: px
Start display at page:

Download "Approximating the Sum of a Convergent Series"

Transcription

1 Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA The BC Calculus Course Descriptio metios how techology ca be used to explore covergece ad divergece of series, ad lists various tests for covergece ad divergece as topics to be covered. But o specific metio is made of actually estimatig the sum of a series, ad the oly discussio of error bouds is for alteratig series ad the Lagrage error boud for Taylor polyomials. With just a little additioal effort, however, studets ca easily approximate the sum of may commo coverget series ad determie how precise that approximatio will be. Approximatig the Sum of a Positive Series Here are two methods for estimatig the sum of a positive series whose covergece has bee established by the itegral test or the ratio test. Some fairly weak additioal requiremets are made o the terms of the series. Proofs are give i the appedix. Let S ad let the th partial sum be S a k.. Suppose f() where the graph of f is positive, decreasig, ad cocave up, ad the improper itegral f(x) dx coverges. The S + + f(x) dx + + k < S < S + f(x) dx +. () (If the coditios for f oly hold for x N, the iequality () would be valid for N.) +. Suppose ( ) is a positive decreasig sequece ad lim L <. If + If + decreases to the limit L, the ( ) L S + L icreases to the limit L, the S < S < S () < S < S + ( ) L. (3) L

2 Example : S The fuctio f(x) is positive with a graph that is decreasig ad cocave up for x, ad x f() for all. I additio, f(x) dx coverges. This series coverges by the itegral test. By iequality (), S ( + ) < S < S + ( + ). (4) This iequality implies that S is cotaied i a iterval of width ( + ) + ( + ). If we wated to estimate S with error less tha 0.000, we could use a value of with < (+) ad the take the average of the two edpoits i iequality (4) as a approximatio for S. The table feature o a graphig calculator shows that 7 is the first value of that works. Iequality (4) the implies that < S < ad a reasoable approximatio would be S.645 to three decimal places. With 00, iequality (4) actually shows that < S < , ad hece we kow for sure that S Of course, i this case we actually kow that S π Notice also that S , so the partial sum with 00 terms is a poor approximatio by itself. Example : S Let f(x) 4 + x. The graph of f is decreasig ad cocave up for x. Also x 4 + x x 4 + π 4 arcta( ) ad so the improper itegral coverges. We ca therefore use iequality () for, ad so S + π 4 arcta(( + ) ) + + (( + ) 4 + ) < S < S + π 4 + arcta( ) (( + ) 4 + ). for. Usig 0 i this iequality yields < S < We ca coclude that S to three decimal places. Example 3: S 0! The terms of this series are decreasig. I additio, + ( + )!! + We will use the covetio for positive edpoits of trucatig the left edpoit of the iterval ad roudig up the right edpoit. This will make the iterval slightly larger tha that give by the actual symbolic iequality.

3 which decreases to the limit L 0. By iequality () S < S < S + (+)! + S +!. for all. Usig 0 i this iequality yields.7888 < S <.7889 ad hece S These, of course, are the first seve decimal places of e Example 4: S We have 5 + ( + ) which icreases to the limit L 5. Accordig to iequality (3) which simplifies to S + S + (+) ( + ( ) + 5 ) < S < S + 5 ( ) 5 < S < S With 5, this iequality shows that < S < Example 5: S We have! + ( ) ( + )! ( + ) +! ( + which is less tha for all ad which decreases to the limit L e. From iequality () we get (after some simplificatio) Usig 0 gives < S < S +! e < S < S! + ( + ). Approximatig the Sum of a Alteratig Series Let S ( ) + ad let the th partial sum be S ( ) k+ a k. We assume that ( ) is a positive decreasig sequece that coverges to 0.. The stadard error boud is give by k S + < S < S + + (5) ) 3

4 . Suppose the sequece defied by b + decreases mootoically to 0. (Oe way to achieve this is if f() where f is positive with a graph that is decreasig asymptotically to 0 ad cocave up.) The if S < S, the S + + < S < S + ; (6) if S < S, the S < S < S +. (7) Both of these ca be summarized by the iequality + < S S <. Iequality (5) is credited to Leibiz ad is the error boud described i the BC Calculus Course Descriptio. Iequalities (6) ad (7) are cosequeces of a proof published i 96 by Philip Calabrese, the a udergraduate studet at the Illiois Istitute of Techology (see referece []). Calabrese proved that S S < ɛ if ɛ, ad that furthermore, if ɛ for some, the S is the first partial sum withi ɛ of the sum S. See the appedix for the derivatio of iequalities (6) ad (7). Example 6: S ( ) + 4 This is a alteratig series that coverges by the alteratig series test. If f(x) 4 x, the the graph of f is positive, decreasig to 0, ad cocave up for x. For odd, iequality (7) implies that S < S < S +. (8) If we wated to estimate the value of S with error less tha 0.000, the typical method usig the error boud from iequality (5) would use a value of for which < This would require usig 0,000 terms. O the basis of iequality (8), however, we ca take as a estimate for S the midpoit of that iterval, that is, for odd, S S ( + + ) S 4 4, (9) with a error less tha half the width of the iterval. So for a error less tha 0.000, we oly eed ( ) + 4 < The first odd solutio is 7, just a bit less tha 0,000! The estimate from (9) usig 7 is S 3.459, with error less tha Sice S π, this estimate is actually withi of the true value. By the way, the partial sum S 7 is approximately

5 Example 7: S ( ) 8 ()! 0 This is a alteratig series that coverges by the alteratig series test. Let b +. It is ot obvious that the sequece b decreases mootoically to 0. A ivestigatio with the table feature of a graphig calculator, however, suggests that this is true for 3. We ca therefore use iequality (6) whe is a odd iteger greater tha 3 (ote that iequality (6) holds for odd s because this series starts with 0.) Hece S ( + )! < S < S + 8 ()! for odd 3. With 9 we ca estimate that S lies i the iterval ( , ), a iterval of legth But wait, we ca actually do better tha this! Sice the terms of this series decrease so quickly because of the factorial i the deomiator, we actually have + < for 3. So if we combie iequalities (5) ad (6), we ca deduce that for this series, 8 + S + ( + )! < S < S + 8+ ( + )! for odd 3. Now 9 gives the iterval ( , ) cotaiig the value of S, a iterval of legth (Note: What is the exact sum of this series?) Refereces [] Bart Brade, Calculatig Sums of Ifiite Series, The America Mathematical Mothly, Vol. 99, No. 7. (Aug. Sep., 99), [] Philip Calabrese, A Note o Alteratig Series, The America Mathematical Mothly, Vol. 69, No. 3. (Mar., 96), 5 7. (Reprited i Selected Papers o Calculus, The Mathematical Associatio of America, 968, ) [3] Rick Kremiski, Usig Simpso s Rule to Approximate Sums of Ifiite Series, The College Mathematics Joural, Vol. 8, No. 5. (Nov., 997), [4] R.K. Morley, The Remaider i Computig by Series, The America Mathematical Mothly, Vol. 57, No. 8. (Oct., 950), [5] R.K. Morley, Further Note o the Remaider i Computig by Series, The America Mathematical Mothly, Vol. 58, No. 6. (Ju. Jul., 95), Appedix Proof of Iequality () Let S ad let S a k. Suppose f() where the graph of f is positive, decreasig k to 0, ad cocave up, ad the improper itegral f(x) dx coverges. The series coverges by the itegral test. Because the graph is cocave up, the area of the shaded trapezoid of width show i Figure () is greater tha the area uder the curve. Therefore For egative edpoits, roud dow the left edpoit ad trucate the right edpoit. 5

6 Figure Figure Hece f(x) dx < (+ + + ). f(x) dx < (+ + + ) + ( ) + ( ) S S + I Figure (), the graph of f lies above that taget lie at x + (because of the positive cocavity) ad therefore also lies above the cotiuatio of the secat lie betwee x + ad x +. This implies that the area of the shaded trapezoid i Figure () of width betwee x ad x + is less tha the area uder the curve, ad so Hece + f(x) dx > + + (+ + ). f(x) dx > + + (+ + ) (+ +3 ) (+3 +4 ) S S Proof of Iequalities () ad (3) Let S ad let S k + a k. Suppose ( ) is a positive decreasig sequece ad lim L <, where the ratios decrease to L. The series coverges by the ratio test. 6

7 Let r + <. The a k+ a k We therefore coclude that S S k+ < r for all k. Hece a k + < r + < + r < r +3 < + r < r 3. +k < k r k r r + + k. But we also have L < a k+ a k for all k. By a similar argumet as above, S S k+ a k +k > k L k L L. Combiig these two results gives iequality (). A similar argumet for the iequalities with r ad L reversed proves iequality (3). Proof of Iequalities (6) ad (7) Let S ( ) + ad let S k k ( ) k+ a k, where ( ) is positive decreasig sequece that coverges to 0. Let b +, where we assume that the sequece (b ) also decreases mootoically to 0. The ad Because the sequece (b ) decreases, S S + ( ) (b + + b +3 + b +5 + ) S S + ( ) + (b + b + + b +4 + ). S S b + + b +3 + b +5 + < b + b + + b +4 + S S. Therefore S S < S S. Similarly, S S + < S S. But S lies betwee the successive partial sums, so it follows that ad Combiig these two results shows that S S S S + S S > S S + S + S S S + + S S < S S. + < S S < from which iequalities (6) ad (7) ca be obtaied. 7

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

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

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

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

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

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

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

MATH 361 Homework 9. Royden Royden Royden

MATH 361 Homework 9. Royden Royden Royden MATH 61 Homework 9 Royde..9 First, we show that for ay subset E of the real umbers, E c + y = E + y) c traslatig the complemet is equivalet to the complemet of the traslated set). Without loss of geerality,

More information

Continued Fractions continued. 3. Best rational approximations

Continued Fractions continued. 3. Best rational approximations Cotiued Fractios cotiued 3. Best ratioal approximatios We hear so much about π beig approximated by 22/7 because o other ratioal umber with deomiator < 7 is closer to π. Evetually 22/7 is defeated by 333/06

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

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

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find 1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

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

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

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

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

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series 8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

More information

TAYLOR SERIES, POWER SERIES

TAYLOR SERIES, POWER SERIES TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the

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

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

a 4 = 4 2 4 = 12. 2. Which of the following sequences converge to zero? n 2 (a) n 2 (b) 2 n x 2 x 2 + 1 = lim x n 2 + 1 = lim x

a 4 = 4 2 4 = 12. 2. Which of the following sequences converge to zero? n 2 (a) n 2 (b) 2 n x 2 x 2 + 1 = lim x n 2 + 1 = lim x 0 INFINITE SERIES 0. Sequeces Preiary Questios. What is a 4 for the sequece a? solutio Substitutig 4 i the expressio for a gives a 4 4 4.. Which of the followig sequeces coverge to zero? a b + solutio

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

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

f(x + T ) = f(x), for all x. The period of the function f(t) is the interval between two successive repetitions.

f(x + T ) = f(x), for all x. The period of the function f(t) is the interval between two successive repetitions. Fourier Series. Itroductio Whe the Frech mathematicia Joseph Fourier (768-83) was tryig to study the flow of heat i a metal plate, he had the idea of expressig the heat source as a ifiite series of sie

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

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

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

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

Homework 1 Solutions

Homework 1 Solutions Homewor 1 Solutios Math 171, Sprig 2010 Please sed correctios to herya@math.staford.edu 2.2. Let h : X Y, g : Y Z, ad f : Z W. Prove that (f g h = f (g h. Solutio. Let x X. Note that ((f g h(x = (f g(h(x

More information

AP Calculus AB 2006 Scoring Guidelines Form B

AP Calculus AB 2006 Scoring Guidelines Form B AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

Holomorphic vector-valued functions Paul Garrett garrett/ 1. Definition, examples

Holomorphic vector-valued functions Paul Garrett  garrett/ 1. Definition, examples (February 9, 2005) Holomorphic vector-valued fuctios Paul Garrett garrett@math.um.edu http://www.math.um.edu/ garrett/ Abstract: Oe of the first goals of a presetatio of classical complex fuctio theory

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

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

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

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

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

Math 113 HW #11 Solutions

Math 113 HW #11 Solutions Math 3 HW # Solutios 5. 4. (a) Estimate the area uder the graph of f(x) = x from x = to x = 4 usig four approximatig rectagles ad right edpoits. Sketch the graph ad the rectagles. Is your estimate a uderestimate

More information

Convexity, Inequalities, and Norms

Convexity, Inequalities, and Norms Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

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

1 n. n > dt. t < n 1 + n=1

1 n. n > dt. t < n 1 + n=1 Math 05 otes C. Pomerace The harmoic sum The harmoic sum is the sum of recirocals of the ositive itegers. We kow from calculus that it diverges, this is usually doe by the itegral test. There s a more

More information

Fourier Series and the Wave Equation Part 2

Fourier Series and the Wave Equation Part 2 Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries

More information

7) an = 7 n 7n. Solve the problem. Answer the question. n=1. Solve the problem. Answer the question. 16) an =

7) an = 7 n 7n. Solve the problem. Answer the question. n=1. Solve the problem. Answer the question. 16) an = Eam Name MULTIPLE CHOICE. Choose the oe alterative that best comletes the statemet or aswers the questio. ) Use series to estimate the itegral's value to withi a error of magitude less tha -.. l( + )d..79.9.77

More information

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern. 5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

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

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

Asymptotic Growth of Functions

Asymptotic Growth of Functions CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

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

1 Notes on Little s Law (l = λw)

1 Notes on Little s Law (l = λw) Copyright c 29 by Karl Sigma Notes o Little s Law (l λw) We cosider here a famous ad very useful law i queueig theory called Little s Law, also kow as l λw, which asserts that the time average umber of

More information

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009) 18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

More information

A Simplified Binet Formula for k-generalized Fibonacci Numbers

A Simplified Binet Formula for k-generalized Fibonacci Numbers A Simplified Biet Formula for k-geeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 dresdeg@wlu.edu Abstract I this paper, we preset a particularly

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

1. Strong vs regular indiction 2. Strong induction examples: ! Divisibility by a prime! Recursion sequence: product of fractions

1. Strong vs regular indiction 2. Strong induction examples: ! Divisibility by a prime! Recursion sequence: product of fractions Today s Topics: CSE 0: Discrete Mathematics for Computer Sciece Prof. Miles Joes 1. Strog vs regular idictio. Strog iductio examples:! Divisibility by a prime! Recursio sequece: product of fractios 3 4

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

Entropy Rates of a Stochastic Process

Entropy Rates of a Stochastic Process Etropy Rates of a Stochastic Process Best Achievable Data Compressio Radu Trîmbiţaş October 2012 1 Etropy Rates of a Stochastic Process Etropy rates The AEP states that H(X) bits suffice o the average

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

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

Confidence Intervals

Confidence Intervals Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more

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

if A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,

if A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S, Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σ-algebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio

More information

Confidence Intervals for One Mean with Tolerance Probability

Confidence Intervals for One Mean with Tolerance Probability Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with

More information

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

Sequences, Series and Convergence with the TI 92. Roger G. Brown Monash University Sequeces, Series ad Covergece with the TI 92. Roger G. Brow Moash Uiversity email: rgbrow@deaki.edu.au Itroductio. Studets erollig i calculus at Moash Uiversity, like may other calculus courses, are itroduced

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

1. C. The formula for the confidence interval for a population mean is: x t, which was

1. C. The formula for the confidence interval for a population mean is: x t, which was s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

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

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

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

Page 2 of 14 = T(-2) + 2 = [ T(-3)+1 ] + 2 Substitute T(-3)+1 for T(-2) = T(-3) + 3 = [ T(-4)+1 ] + 3 Substitute T(-4)+1 for T(-3) = T(-4) + 4 After i

Page 2 of 14 = T(-2) + 2 = [ T(-3)+1 ] + 2 Substitute T(-3)+1 for T(-2) = T(-3) + 3 = [ T(-4)+1 ] + 3 Substitute T(-4)+1 for T(-3) = T(-4) + 4 After i Page 1 of 14 Search C455 Chapter 4 - Recursio Tree Documet last modified: 02/09/2012 18:42:34 Uses: Use recursio tree to determie a good asymptotic boud o the recurrece T() = Sum the costs withi each level

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

Chapter Suppose you wish to use the Principle of Mathematical Induction to prove that 1 1! + 2 2! + 3 3! n n! = (n + 1)! 1 for all n 1.

Chapter Suppose you wish to use the Principle of Mathematical Induction to prove that 1 1! + 2 2! + 3 3! n n! = (n + 1)! 1 for all n 1. Chapter 4. Suppose you wish to prove that the followig is true for all positive itegers by usig the Priciple of Mathematical Iductio: + 3 + 5 +... + ( ) =. (a) Write P() (b) Write P(7) (c) Write P(73)

More information

Math 152 Final Exam Review

Math 152 Final Exam Review Math 5 Fial Eam Review Problems Math 5 Fial Eam Review Problems appearig o your i-class fial will be similar to those here but will have umbers ad fuctios chaged. Here is a eample of the way problems selected

More information

when n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.

when n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on. Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have

More information

x(x 1)(x 2)... (x k + 1) = [x] k n+m 1

x(x 1)(x 2)... (x k + 1) = [x] k n+m 1 1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,

More information

2-3 The Remainder and Factor Theorems

2-3 The Remainder and Factor Theorems - The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

More information

Using Excel to Construct Confidence Intervals

Using Excel to Construct Confidence Intervals OPIM 303 Statistics Ja Stallaert Usig Excel to Costruct Cofidece Itervals This hadout explais how to costruct cofidece itervals i Excel for the followig cases: 1. Cofidece Itervals for the mea of a populatio

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

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized? 5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso

More information

Winter Camp 2012 Sequences Alexander Remorov. Sequences. Alexander Remorov

Winter Camp 2012 Sequences Alexander Remorov. Sequences. Alexander Remorov Witer Camp 202 Sequeces Alexader Remorov Sequeces Alexader Remorov alexaderrem@gmail.com Warm-up Problem : Give a positive iteger, cosider a sequece of real umbers a 0, a,..., a defied as a 0 = 2 ad =

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

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

Ma/CS 6b Class 17: Extremal Graph Theory

Ma/CS 6b Class 17: Extremal Graph Theory //06 Ma/CS 6b Class 7: Extremal Graph Theory Paul Turá By Adam Sheffer Extremal Graph Theory The subfield of extremal graph theory deals with questios of the form: What is the maximum umber of edges that

More information

Review for 1 sample CI Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Review for 1 sample CI Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Review for 1 sample CI Name MULTIPLE CHOICE. Choose the oe alterative that best completes the statemet or aswers the questio. Fid the margi of error for the give cofidece iterval. 1) A survey foud that

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

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 LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE

THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE JAVIER CILLERUELO Abstract. We obtai, for ay irreducible quadratic olyomial f(x = ax 2 + bx + c, the asymtotic estimate log l.c.m. {f(1,..., f(} log. Whe

More information

Tangent circles in the ratio 2 : 1. Hiroshi Okumura and Masayuki Watanabe. In this article we consider the following old Japanese geometry problem

Tangent circles in the ratio 2 : 1. Hiroshi Okumura and Masayuki Watanabe. In this article we consider the following old Japanese geometry problem 116 Taget circles i the ratio 2 : 1 Hiroshi Okumura ad Masayuki Wataabe I this article we cosider the followig old Japaese geometry problem (see Figure 1), whose statemet i [1, p. 39] is missig the coditio

More information

0,1 is an accumulation

0,1 is an accumulation Sectio 5.4 1 Accumulatio Poits Sectio 5.4 Bolzao-Weierstrass ad Heie-Borel Theorems Purpose of Sectio: To itroduce the cocept of a accumulatio poit of a set, ad state ad prove two major theorems of real

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

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

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

Factors of sums of powers of binomial coefficients

Factors of sums of powers of binomial coefficients ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the

More information

Measurable Functions

Measurable Functions Measurable Fuctios Dug Le 1 1 Defiitio It is ecessary to determie the class of fuctios that will be cosidered for the Lebesgue itegratio. We wat to guaratee that the sets which arise whe workig with these

More information

Taylor Series and Polynomials

Taylor Series and Polynomials Taylor Series ad Polyomials Motivatios The purpose of Taylor series is to approimate a fuctio with a polyomial; ot oly we wat to be able to approimate, but we also wat to kow how good the approimatio is.

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

8 The Poisson Distribution

8 The Poisson Distribution 8 The Poisso Distributio Let X biomial, p ). Recall that this meas that X has pmf ) p,p k) p k k p ) k for k 0,,...,. ) Agai, thik of X as the umber of successes i a series of idepedet experimets, each

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

3. Greatest Common Divisor - Least Common Multiple

3. Greatest Common Divisor - Least Common Multiple 3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

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