# ORDERS OF GROWTH KEITH CONRAD

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 such as covergece of improper itegrals ad ifiite series We wat to compare the growth of three differet kids of fuctios of x, as x : power fuctios for r > (such as x 2 or x = x /2 ), expoetial fuctios a x for a >, logarithmic fuctios log b x for b > Examples are plotted i Figure The relative sizes are differet for x ear ad for large x f(x) x Figure Graph of y = x 2, y = x, y = e x, y = l(x) All power fuctios, expoetial fuctios, ad logarithmic fuctios (as defied above) ted to as x But these three classes of fuctios ted to at differet rates The mai result we wat to focus o is the followig oe It says e x grows faster tha ay power fuctio while log x grows slower tha ay power fuctio (A power fuctio meas with r >, so /x 2 = x 2 does t cout)

2 2 KEITH CONRAD log x Theorem For each r >, lim = ad lim x ex x = This is illustrated i Figure The fuctios icrease at first, but ted to for larger x After we prove Theorem ad look at some cosequeces of it i Sectio 2, we will compare power, expoetial, ad log fuctios with the sequeces! ad i Sectio 3 At the ed we will show that betwee ay two sequeces with differet orders of growth we ca isert ifiitely may sequeces with differet orders of growth i betwee them 2 Proof of Theorem ad some corollaries Proof (of Theorem ) First we focus o the limit /e x Whe r = this says x (2) as x ex This result follows from L Hopital s rule To derive the geeral case from this special case, write ( ) x/r r (22) e x = rr e x/r With r stayig fixed, as x also x/r, so (x/r)/e x/r by (2) with x/r i place of x The the right side of (22) teds to as x, so we re doe Now we show (log x)/ as x Writig y for log( ) = r log x, log x = y/r e y = r y e y As x, also y, so (/r)(y/e y ) by (2) with y i place of x Thus (log x)/ p(x) Corollary 2 For ay polyomial p(x), lim x e x = Proof By Theorem, for ay k > we have x k /e x as x This is also true whe k = Writig p(x) = a d x d + a d x d + + a x + a, we have p(x) x d e x = a d e x + a x d x d e x + + a e x + a e x Each x k /e x appearig here teds to as x, so p(x)/e x teds to as x (log x) k Corollary 22 For ay r > ad k >, lim x = Proof Let y = log( ) = r log x, so (log x) k = yk /r k e y = r k yk e y As x, also y Therefore (/r k )(y k /e y ) by Theorem (sice r k > ) We derived Corollary 22 from Theorem, but the argumet ca be reversed too Take k = i Corollary 22 to get the log part of Theorem ad use the chage of variables y = e x i /e x to get the expoetial part of Theorem from Corollary 22 Specifically, whe y = e x (log y)r =, ex y ad as x we have y = e x, so by Corollary 22 we get (log y) r /y /e x as x Therefore

3 ORDERS OF GROWTH 3 (log x) k Corollary 23 For ay o-costat polyomial p(x) ad positive umber k, lim x p(x) Proof For large x, p(x) sice o-zero polyomials have oly a fiite umber of roots Write p(x) = a d x d + a d x d + + a x + a, where d > ad a d The (log x) k p(x) = (log x)k x d a d + a d /x + + a /x d As x, the first factor teds to by Corollary 22 while the secod factor teds to /a d, so the product teds to Corollary 24 As x, x /x Proof The logarithm of x /x is log(x /x ) = (log x)/x, which teds to as x Expoetiatig, x /x = e (log x)/x e = We coclude this sectio by givig a secod proof of Corollary 22 which does t rely o aythig we have doe so far Thus, we could cosider Corollary 22 as the mai result ad Theorem as a special case! Proof We will use estimates o the itegral for log x: log x = x dt t For r >, we have /t /t r whe t Therefore whe x > < log x = x dt t x dt t r = x t r dt = xr r r < xr r If we ru through these estimates with r/2 i place of r (which is fie sice r/2 > too) the we get (23) < log x xr/2 r/2 = 2xr/2 r The reaso we use r/2 is because ow whe we divide by we get a decayig term o the right side: < log x 2 r/2 As x, the right side teds to, so (log x)/ But this is t complete: we wat (log x) k / for ay k >, ot just (log x)/ To get this, let s ru through the iequalities agai usig r/(2k) i place of r This amouts to substitutig r/2 with r/(2k) i (23), ad the result is = Now raise to the k-th power: < log x xr/(2k) r/(2k) = 2kxr/(2k) r < (log x) k ( ) 2k k /2 r

4 4 KEITH CONRAD Dividig by as we did before, < (log x)k ( ) 2k k r /2 As x, the right side teds to, so we have show (log x) k / Replacig e x with a x for ay a > ad log x with log b x for ay b > leads to completely aalogous results Theorem 25 Fieal umbers a > ad b > For ay r > ad iteger k >, lim x a x =, For ay o-costat polyomial p(x), lim x lim (log b x) k x = p(x) (log =, lim b x) k ax x p(x) Proof To deduce this theorem from earlier results, write a x = e (log a)x ad log b x = (log x)/(log b) The umbers log a ad log b are positive The, for istace, if we set y = (log a)x, a x = xr e (log a)x = y r (log a) r e y Whe x, also y sice log a >, so the behavior of /a x follows from that of y r /e y usig Theorem Sice log b x = (log x)/(log b) is a costat multiple of log x, carryig over the results o log x to log b x is just a matter of rescalig For istace, if we set y = log x, so log b x = y/ log b, the (log b x) k = yk /(log b) k e ry = (log b) k = y k (e r ) y As x, also y, so the expoetial fuctio (e r ) y domiates over the power fuctio y k : y k /(e r ) y Therefore (log b x) k / as x 3 Growth of basic sequeces We wat to compare the growth of five kids of sequeces: power sequeces r for r > :, 2 r, 3 r, 4 r, 5 r, expoetial sequeces a for a > : a, a 2, a 3, a 4, a 5, log sequeces log b for b > :, log b 2, log b 3, log b 4, log b 5,!:, 2, 6, 24, 2, :, 4, 27, 256, 325, The first three sequeces are just the fuctios we have already treated, except the real variable x has bee replaced by a iteger variable That is, we are lookig at those old fuctios at iteger values of x ow Some otatio to covey domiatig rates of growth will be coveiet For two sequeces x ad y, write x y to mea x /y as I other words, x grows substatially slower tha y (if it just grew at half the rate, for istace, the x /y would be aroud /2 rather tha ted to ) For istace, 2 ad

5 ORDERS OF GROWTH 5 Remark 3 The otatio x y does ot mea x < y for all Maybe some iitial terms i the x sequece are larger tha the correspodig oes i the y sequece, but this will evetually stop ad the log term growth of y domiates For istace, 2 eve though 2 < for all small Ideed, the ratio 2 = teds to as, but the ratio is ot small util gets quite large Theorem tells us that (3) log r e for ay r > By Theorem 25, we ca say more geerally that (32) log b r a for ay a > ad b > How do the sequeces! ad fit ito (32)? They belog o the right, as follows Theorem 32 For ay a >, a! Equivaletly, a! lim =, lim! = Proof To compare a ad!, we will use Euler s beautiful itegral formula for!: (33)! = x e x dx I case you re ot familiar with (33), let s recall how to prove it by iductio o Whe =, x e x dx = e x dx, which is by itegratio Assumig x e x dx =! for some, we will compute x + e x dx usig itegratio by parts with u = x + ad dv = e x dx: x + e x dx = x + e x + ( + )x e x dx = lim b b+ e b + ( + )x e x dx The limit is by Theorem 25, ad the itegral is ( + ) x e x dx, which by iductio is ( + )! = ( + )! To apply (33), we obtai a a lower boud for! by makig the itegral ru over [, ]:! > x e x dx O the iterval [, ], e x has its smallest value at the right ed: e x e Therefore x e x x e o [, ] Itegratig both sides of this iequality from x = to x = gives x e x dx = e x e dx x dx = + e + ( ) = e +

6 6 KEITH CONRAD ( ) Therefore! > e +, so a! < a ( ae (/e) (/( + )) = ) + This fial expressio is a upper boud o a /! How does it behave as? For large, ae/ /2, so (ae/) (/2) Therefore (ae/) Sice the other factor ( + )/ teds to, we see our upper boud o a /! teds to, so a /! as To show the other part of the theorem, that!/ as, we will get a upper boud o! ad divide the upper boud by Write e x as e x/2 e x/2 i Euler s factorial itegral:! = x e x dx = (x e x/2 )e x/2 dx The fuctio x e x/2 drops off to as x Where does it have its maximum value? The derivative is x e x/2 ( x/2) (check this), so x e x/2 vaishes at x = 2 Checkig the sigs of the derivative to the left ad right of x = 2, we see x e x/2 has a maximum value at x = 2, where the value is (2) e Therefore x e x/2 (2) e for all x >, so! = (x e x/2 )e x/2 dx (2) e e x/2 dx = (2) e e x/2 dx = (2) e 2 Dividig throughout by gives ( )! 2 2 e Sice 2 < e, the right side teds to, so!/ as The fact that a! is ituitively reasoable, for the followig reaso: each of these expressios (a,!, ad ) is a product of umbers, but the ature of these umbers is differet I a, all umbers are the same value a, which is idepedet of : a = a } a {{ a} times I!, the umbers are the itegers from to :! = 2 3 ( ) Sice the terms i this product keep growig, while the terms i a stay the same, it makes sese that! grows faster tha a (at least oce gets larger tha a) I, all umbers equal : = } {{ } times Sice all the terms i this product equal, while i! the terms are the umbers from to, it is plausible that grows a lot faster tha! To summarize our results o sequeces, we combie (32) ad Theorem 32: log b r a!

7 ORDERS OF GROWTH 7 Here a >, b >, ad r > (ot just r >!) All sequeces here ted to as, but the rates of growth are all differet: ay sequece which comes to the left of aother sequece o this list grows at a substatially smaller rate, i the sese that the ratio teds to For example, ca we fid a (atural) sequece whose growth is itermediate betwee ad r for every r >? That is, we wat to fid a sigle sequece of umbers x such that x r for every r > Oe choice is x = log Ideed, log = log, so log, ad for ay r > log r = log r, which teds to sice r > ad log grows slower tha ay power fuctio (with a positive expoet) by Theorem Usig powers of log, we ca write dow ifiitely may sequeces with differet rates of growth betwee ad every sequece r for r > : log (log ) 2 (log ) 3 (log ) k r, where k rus through the positive itegers Is it possible to isert ifiitely may sequeces with differet rates of growth betwee ay two sequeces with differet rates of growth? Theorem 33 If x y, there are sequeces {z () }, {z (2) }, {z (3) }, such that x z () z (2) z (3) y? Proof Sice x /y, for large the ratio x /y is small Specifically, < x /y < for large For small positive umbers, takig roots makes them larger but less tha : < a < = < a < a < 3 a < < k a < < Sice x /y < for large, this presets us with the iequalities < x x x x < < y y 3 < < y k < < y for large ad k =, 2, 3, Multiply through by y : (34) < x < x y < x /3 y 2/3 < < x /k y /k < < y For k < l, the ratio of the k-th root sequece to the l-th root sequece is x /k y /k ( ) /k /l x x /l y /l = y Sice /k /l >, this ratio teds to as Therefore (34) leads to ifiitely may sequeces with growth itermediate betwee {x } ad {y }, amely the sequeces z (k) = x /k y /k for k = 2, 3, 4, : (35) x x y x /3 y 2/3 x /k (If you wat to label the first sequece with k =, set z (k) y /k y = x /(k+) y /(k+) for k =, 2, 3, )

8 8 KEITH CONRAD The differece betwee (34) ad (35) is that (34) is a set of iequalities which is valid for large (amely large eough to have x /y < ), while (35) is a statemet about rates of growth betwee differet sequeces: it makes o sese to ask if (35) is true at a particular value of, ay more tha it would make sese to ask if the limit relatio + is true at = 45

### 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

### 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

### 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

### 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

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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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,

### 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.,

### 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...

### The Euler Totient, the Möbius and the Divisor Functions

The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship

### 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

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

### 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,...,

### .04. This means \$1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

### 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

### Factoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett <garrett@math.umn.edu>

(March 16, 004) Factorig x 1: cyclotomic ad Aurifeuillia polyomials Paul Garrett Polyomials of the form x 1, x 3 1, x 4 1 have at least oe systematic factorizatio x 1 = (x 1)(x 1

### 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.

### 7. Sample Covariance and Correlation

1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

### 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

### 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

### Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015

CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a

### 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

### 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.

### Class Meeting # 16: The Fourier Transform on R n

MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,

### 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

### 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

### Recursion and Recurrences

Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,

### 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

### 0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

### Lecture 5: Span, linear independence, bases, and dimension

Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;

### Overview of some probability distributions.

Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

### 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

### Chapter One BASIC MATHEMATICAL TOOLS

Chapter Oe BAIC MATHEMATICAL TOOL As the reader will see, the study of the time value of moey ivolves substatial use of variables ad umbers that are raised to a power. The power to which a variable is

### 1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

### Limits, Continuity and derivatives (Stewart Ch. 2) say: the limit of f(x) equals L

Limits, Cotiuity ad derivatives (Stewart Ch. 2) f(x) = L say: the it of f(x) equals L as x approaches a The values of f(x) ca be as close to L as we like by takig x sufficietly close to a, but x a. If

### 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

### 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

### Distributions of Order Statistics

Chapter 2 Distributios of Order Statistics We give some importat formulae for distributios of order statistics. For example, where F k: (x)=p{x k, x} = I F(x) (k, k + 1), I x (a,b)= 1 x t a 1 (1 t) b 1

### 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

### A probabilistic proof of a binomial identity

A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

### Solving Divide-and-Conquer Recurrences

Solvig Divide-ad-Coquer Recurreces Victor Adamchik A divide-ad-coquer algorithm cosists of three steps: dividig a problem ito smaller subproblems solvig (recursively) each subproblem the combiig solutios

### Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

### FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if

### Hypothesis testing. Null and alternative hypotheses

Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

### 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.

### CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

### 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

### 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

### NUMBERS COMMON TO TWO POLYGONAL SEQUENCES

NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively

### WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This

### THE UNLIKELY UNION OF PARTITIONS AND DIVISORS

THE UNLIKELY UNION OF PARTITIONS AND DIVISORS Abdulkadir Hasse, Thomas J. Osler, Mathematics Departmet ad Tirupathi R. Chadrupatla, Mechaical Egieerig Rowa Uiversity Glassboro, NJ 828 I the multiplicative

### 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

### 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

### Lecture 4: Cheeger s Inequality

Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a d-regular

### Incremental calculation of weighted mean and variance

Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

### 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

### 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

### Linear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant

MTH6140 Liear Algebra II Notes 4 1st November 2010 4 Determiats The determiat is a fuctio defied o square matrices; its value is a scalar. It has some very importat properties: perhaps most importat is

### 8.1 Arithmetic Sequences

MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first

### Section 8.3 : De Moivre s Theorem and Applications

The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =

### Output Analysis (2, Chapters 10 &11 Law)

B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

### 3. Covariance and Correlation

Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics

### THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

### Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis

Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) are

### Chapter 7 Methods of Finding Estimators

Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

### NOTES ON INEQUALITIES FELIX LAZEBNIK

NOTES ON INEQUALITIES FELIX LAZEBNIK Order ad iequalities are fudametal otios of moder mathematics. Calculus ad Aalysis deped heavily o them, ad properties of iequalities provide the mai tool for developig

### 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

### CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

### Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected

### Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites

Gregory Carey, 1998 Liear Trasformatios & Composites - 1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio

### Solutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork

Solutios to Selected Problems I: Patter Classificatio by Duda, Hart, Stork Joh L. Weatherwax February 4, 008 Problem Solutios Chapter Bayesia Decisio Theory Problem radomized rules Part a: Let Rx be the

### 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

### Metric, Normed, and Topological Spaces

Chapter 13 Metric, Normed, ad Topological Spaces A metric space is a set X that has a otio of the distace d(x, y) betwee every pair of poits x, y X. A fudametal example is R with the absolute-value metric

### Running Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis

Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.

### Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2

74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is

### 1 Introduction to reducing variance in Monte Carlo simulations

Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by

### Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

### 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

### 1 Correlation and Regression Analysis

1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

### THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average

### THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

### Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:

Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network

### Exponential function: For a > 0, the exponential function with base a is defined by. f(x) = a x

MATH 11011 EXPONENTIAL FUNCTIONS KSU AND THEIR APPLICATIONS Defiitios: Expoetial fuctio: For a > 0, the expoetial fuctio with base a is defied by fx) = a x Horizotal asymptote: The lie y = c is a horizotal

### Ekkehart Schlicht: Economic Surplus and Derived Demand

Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/

### MARTINGALES AND A BASIC APPLICATION

MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this

### 1.3 Binomial Coefficients

18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to

### GCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4

GCE Further Mathematics (660) Further Pure Uit (MFP) Tetbook Versio: 4 MFP Tetbook A-level Further Mathematics 660 Further Pure : Cotets Chapter : Comple umbers 4 Itroductio 5 The geeral comple umber 5

### 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

### Section 1.6: Proof by Mathematical Induction

Sectio.6 Proof by Iductio Sectio.6: Proof by Mathematical Iductio Purpose of Sectio: To itroduce the Priciple of Mathematical Iductio, both weak ad the strog versios, ad show how certai types of theorems

### 5 Boolean Decision Trees (February 11)

5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

### Irreducible polynomials with consecutive zero coefficients

Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem