SUMS OF GENERALIZED HARMONIC SERIES. Michael E. Ho man Department of Mathematics, U. S. Naval Academy, Annapolis, Maryland

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "SUMS OF GENERALIZED HARMONIC SERIES. Michael E. Ho man Department of Mathematics, U. S. Naval Academy, Annapolis, Maryland"

Transcription

1 #A46 INTEGERS 4 (204) SUMS OF GENERALIZED HARMONIC SERIES Michael E. Ho ma Departmet of Mathematics, U. S. Naval Academy, Aapolis, Marylad Courtey Moe Departmet of Mathematics, U. S. Naval Academy, Aapolis, Marylad Received: 9/3/3, Accepted: 6/2/4, Published: 8/27/4 Abstract We prove two results o sums of geeralized harmoic series.. Itroductio For oegative itegers a, a 2,..., a k with a a k 2, defie H(a, a 2..., a k ) a ( ) a2 ( k ) a. () k It is easy to show that this series coverges, ad i fact (by cosiderig the partialfractios decompositio of a ( ) a2 ( k ) a k ) that it ca be writte as a ratioal umber plus a sum of values (m) of the Riema zeta fuctio for 2 apple m apple max{a,..., a k }. For example, H(2, 3) 3 apple 2 ( ) 3 apple ( ) 2 ( ) ( ) 2 ( ) 3 63 (2) (3). We call the uatity () a H-series of legth k ad weight a a k. I 2 below we establish various formulas for H-series, i which Stirlig umbers of the first kid are promietly ivolved. These are used i 3 to prove the followig result.

2 INTEGERS: 4 (204) 2 Theorem. For ay positive itegers k, m with m 2, H(a,..., a k ) k (m), a a 2 a k m where the sum is over k-tuples of oegative itegers. The H-series ca be put ito the framework of Shitai zeta fuctios [5]. For a m k matrix A (a i ) ad a k-vector ~x (x,..., x k ), the Shitai zeta fuctio of (s,..., s k ) is (s,..., s k ; A; ~x),..., m 0 (x P m i a i i ) s (x k P m i a. (2) ik i ) s k The a H-series is ust the special case of (2) with m, A [ ], ad ~x (, 2,..., k). Several other special cases of Shitai zeta fuctios have bee extesively studied. These iclude the Bares zeta fuctios [], which is the case k ; ad the multiple zeta values [4, 7], which is the case where m k, A is the k k matrix with s o ad below the mai diagoal ad 0 s above, ad ~x (k, k,..., ). I the latter case the series (2) becomes (s,..., s k ),..., k 0 (k P k i i) s (k P k i2 i) s2 ( k ) s k m s ms2 2. ms k k m >m 2> >m k I fact, Theorem is superficially similar to the famous sum coecture for multiple zeta values, i.e., (a, a 2,..., a k ) (m) (3) a a k m, a i, a > for m 2. The multiple zeta values sum coecture (3) was made by the secod author (see [4]), ad was proved i full geerality by A. Graville [2]. Note, however, that (a,..., a k ) is a k-fold sum while H(a,..., a k ) is a sigle sum. Also, the sum o the left-had side of (3) is over k-tuples of positive itegers rather tha oegative itegers. It is also atural to ask about the sum of all H-series H(a,..., a k ) of weight m over all k-tuples (a,..., a k ) of positive itegers of fixed weight. As idicated by the example H(, ) ( ), the aswer is a ratioal umber. More precisely, we have the followig result, which follows easily from the machiery developed i 2.

3 INTEGERS: 4 (204) 3 Theorem 2. For m k 2, a a k m, a i H(a,..., a k ) (k )! k 2 k 2 i i0 ( ) i (i ) m k. 2. H-Series ad Stirlig Numbers of the First Kid Heceforth we assume m to be a positive iteger greater tha. followig result. We have the Lemma. Let apple i < apple p, ad let a,..., a i, a i,..., a, a,..., a p be fixed oegative itegers. The m k H(a,..., a i, k, a i,..., a, m k, a,..., a p ) i [H(a,..., a i, m, a i,..., a, 0, a,..., a p ) H(a,..., a i, 0, a i,..., a, m, a,..., a p )]. Proof. Evidetly so that ( i )( ) i apple i, H(a,..., a i, k, a i,..., a, m k, a,..., a p ) i [H(a,..., a i, k, a i,..., a, m k, a,..., a p ) Now sum o k to obtai the coclusio. H(a,..., a i, k, a i,..., a, m k, a,..., a p )]. Usig the precedig result, we ca obtai a formula for the sum of all H-series of fixed weight ad legth havig ozero etries at specified locatios. Lemma 2. For k ad apple i 0 < i < < i k apple, a i0 a i a ik m, a 6 0 i i for some k H(a,..., a ) ( ) H (m k) i 0,i (i i 0 )(i i ) (i i )(i i ) (i k i ), (4)

4 INTEGERS: 4 (204) 4 where for p apple H () p, p. Proof. For coveiece, deote by S(i 0, i,..., i k ; m) the left-had side of euatio (4). We proceed by iductio o k, usig Lemma. First, ote that the case k of the result, i.e., ) H(m i S(i 0, i ; m) 0,i, i i 0 follows immediately from Lemma. Now suppose the result holds for k, ad cosider S(i 0,..., i k, i k ; m), i.e., the sum of all H(a,..., a ) with ozero etries a i0, a i,..., a ik addig up to m. Fix a i0,..., a ik, ad sum the H(a,..., a ) from a ik to a ik m a i0 a ik, keepig the sum of a ik ad a ik eual to m a i0 a ik. By Lemma this gives us, after summatio o a i0,..., a ik, i k i k [S(i 0,..., i k, i k ; m ) S(i 0,..., i k, i k ; m )]. (5) By the iductio hypothesis, S(i 0,..., i k ; m ) is k ( ) (m k ) H i 0,i (i i 0 ) (i i )(i i ) (i k i ) ( )k H ad S(i 0,..., i k, i k ; m ) is k ( ) (m k ) H i 0,i (i i 0 ) (i i )(i i ) (i k i )(i k i ) Hece the uatity (5) is k ( ) (m k ) H i 0,i (i i 0 ) (i i )(i i ) (i k i ) (m k ) i 0,i k (i k i 0 ) (i k i k ) ( ) k (m k ) H i 0,i k (i k i 0 ) (i k i k ). i k i i k i k i k ( ) k (m k ) H i 0,i k (i k i 0 ) (i k i k )(i k i k ) ( ) k H (i k i 0 ) (i k i k )(i k i k ) k i (m k ) i 0,i k ( ) H (m k ) i 0,i (i i 0 ) (i i )(i i ) (i k i ).

5 INTEGERS: 4 (204) 5 For 0 apple k apple, let C(k, ; m) be the sum of all H(a,..., a ) of weight m (m k) with exactly k of the a i ozero. Sice each of the H i 0,i i the precedig result is a sum of uatities with apple i p m k 0 apple p apple i apple, for k we ca write k, C(k, ; m) m k (6) where k, property. is ratioal. The the umbers c() k, Lemma 3. For apple k, apple, ( )k k,. have a symmetry/atisymmetry Proof. Borrowig the otatio used i the proof of Lemma 2, we have C(k, ; m) S(i 0, i..., i k ; m). applei 0<i < <i k apple Now ote that Lemma 2 implies that if the S(i 0, i,..., i k ; m) p p 2 2 m k p ( ) m k, S( i k, i k,..., i 0 ; m) ( ) k p p 2 2 m k p ( ) m k. The first (ad last) colums of the umbers k, ca be writte i terms of (usiged) Stirlig umbers of the first kid: we write k for the umber of permutatios of {, 2..., } with exactly k disoit cycles. This otatio follows [3], which is also a useful referece o these umbers. Lemma 4. For apple k apple, k, ( )k ( )! Proof. It su ces to prove the formula for Lemma 3. Now Lemma 2 says that applei 0<i < <i k < apple k., as that for c() k, the follows from ( ) k ( i 0 )( i ) ( i k ),

6 INTEGERS: 4 (204) 6 so to prove the result we eed to show applei 0<i < <i k < ( )! ( i 0 )( i ) ( i k ) apple k. The left-had side is evidetly the sum of all products of k distict factors from the set {, 2,..., }, ad that this is the Stirlig umber k follows from cosideratio of the geeratig fuctio x(x ) (x ) k apple k xk,. (7) The precedig result geeralizes as follows. Lemma 5. For apple k, apple, k, Proof. We eed to show that k, p k, k2 p ( ) p p ( )!( )! p k2 p ( ) p p ( )!( )!. (8). (9) Suppose first that apple k. Usig Lemma 2, all terms that cotribute to cotribute to are of the forms, ad the terms that cotribute to c() (m k) also ad ot to c() ( ) k H i 0, ( i 0 ) ( i k ), i 0 < i < < i k <, (m k) ( ) k 2 H i 0, ( i 0 ) ( i k 2 )(i k ), i 0 < i < < i k 2 < < i k,... (m k) ( ) k H i 0, ( i 0 ) ( i k )(i k 2 ) (i k ), i 0 < i < < i k < < i k 2 < < i k (where we assume throughout that i 0,..., i k are itegers betwee ad ), which

7 INTEGERS: 4 (204) 7 we refer to as classes, 2,.... The cotributio from class is i 0< <i k < The cotributio from class 2 is i <i k k s s ( ) k ( i 0 )( i ) ( i k ) ( ) k k ( )! ( )k k ( )! ( )!. i 0< <i k 2 < i 0< <i k 2 < ( ) k 2 ( i 0 ) ( i k 2 ) ( ) k 2 ( i 0 ) ( i k 2 ) ( ) k ( ) 2 k ( )! ( )!, ad so forth. Note that for class we must have i k 2 2, i k 3 3,..., i k ad the cotributio from this class is i 0< <i k < ( ) k ( i 0 )( i ) ( i k )( )! ( ) ( k ) k 2 ( )! ( )!. Addig together the cotributios, we see that 2 ( ) k k k ( ) ( )!( )! is k 2, (0) ad euatio (9) follows from Lemma 3. Now if > k, the classes k, k 2,..., are evidetly empty. correspodig terms i (0) are zero, except for ( ) ( )k k k k ( )!( )! ( )!. All the This uatity is accouted for by terms that cotribute to ad ot to

8 INTEGERS: 4 (204) 8, provided <i < <i k p k ( ) p (i p )(i p i ) (i p i p )(i p i p ) (i k i p ) k ( )!. () By euatig the expressios give for c () k, by Lemmas 2 ad 4 respectively, we have k <i < <i k apple p from which euatio () follows. ( ) p (i p )(i p i ) (i p i p )(i p i p ) (i k i p ) k ( )!, 3. Proofs of the Theorems First we eed oe more lemma. Lemma 6. For positive itegers ad, Proof. We use the sake oil method of H. Wilf [6]. Let F (x) The F (x) ca be rewritte ( ) ad euatio (2) follows. x ( ). (2) ( ) x x ( x) x ( x) x ( x) ( x) Now we prove Theorems ad 2. ( ). ( ) m p0 p m x m ( x) p x ( x) 2 x 2x2 3x 3,

9 INTEGERS: 4 (204) 9 Proof of Theorem. We have Now H(a,..., a ) C(k, ; m). a a 2 a m k0 ad C(0, ; m) H(m, 0, 0..., 0) H(0, m, 0,..., 0) H(0,..., 0, m) (m) (m) (m) 2 m ( ) m 2 (m) ( ) 2 m ( ) m so to prove the result we eed for all apple apple C(k, ; m) i k, m k, k, i i,. By Lemma 5 this meas we must show ( i )p p i2 p. (3) ( )!( )! i p We rewrite the left-had side of euatio (3) as ( ) p p apple ( )! ( )! i 2 p i. (4) p If p, the ier sum i (4) is apple i i i apple i i2 where we have used euatio (7). Hece sice i i i apple i ( )! i ( )! ( )! ( )!. If p 2, the the ier sum i (4) is apple i 2 p i apple i i i apple ip 2 p 2 ( )! p ( )!, ( )!!,

10 INTEGERS: 4 (204) 0 agai usig euatio (7), from which follows ( )! apple i i 2 p i p. Thus, the sum (4) ca be writte apple 2 p2 If we use euatio (7), this becomes ( ) p p ( )! ( )! 2 The euatio (3) follows from Lemma 6. p apple ( ) p ( )! p. p2 apple ( ) ( ) 2 ( ) apple ( ) ( ). Proof of Theorem 2. It su ces to show that, ( ) 2. ( )! From Lemma 5 we have, p p ( ) p p ( )!( )! ( )! ( ) ( )!( )! ( ) ( ) 2. ( )!

11 INTEGERS: 4 (204) Refereces [] E. W. Bares, O the theory of multiple gamma fuctios, Tras. Cambridge Philos. Soc. 9 (904), [2] A. Graville, A decompositio of Riema s zeta-fuctio, i Aalytic Number Theory, Y. Motohashi (ed.), Lodo Math. Soc. Lect. Note Ser. 247, Cambridge Uiversity Press, New York, 997, pp [3] R. Graham, D. Kuth, ad O. Patashik, Cocrete Mathematics, 2d ed., Addiso-Wesley, New York, 994. [4] M. E. Ho ma, Multiple harmoic series, Pacific J. Math. 52 (992), [5] T. Shitai, O evaluatio of zeta fuctios of totally real algebraic umber fields at opositive itegers, J. Fac. Sci. Uiv. Tokyo Sect. IA Math. 23 (976), [6] H. Wilf, Geeratigfuctioology, 3rd ed., A K Peters, Wellesley, MA, [7] D. Zagier, Values of zeta fuctios ad their applicatios, i First Europea Cogress of Mathematics (Paris, 992), Vol. II, A. Joseph et. al. (eds.), Birkhäuser, Basel, 994, pp

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

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

1. MATHEMATICAL INDUCTION

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

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

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

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

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

{{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

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

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

CS103X: Discrete Structures Homework 4 Solutions

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

More information

A probabilistic proof of a binomial identity

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

More information

A Note on Sums of Greatest (Least) Prime Factors

A Note on Sums of Greatest (Least) Prime Factors It. J. Cotemp. Math. Scieces, Vol. 8, 203, o. 9, 423-432 HIKARI Ltd, www.m-hikari.com A Note o Sums of Greatest (Least Prime Factors Rafael Jakimczuk Divisio Matemática, Uiversidad Nacioal de Luá Bueos

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

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

The Euler Totient, the Möbius and the Divisor Functions

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

More information

Chapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity

More information

8.1 Arithmetic Sequences

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

More information

The Field Q of Rational Numbers

The Field Q of Rational Numbers Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees

More information

1.3 Binomial Coefficients

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

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

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

Permutations, the Parity Theorem, and Determinants

Permutations, the Parity Theorem, and Determinants 1 Permutatios, the Parity Theorem, ad Determiats Joh A. Guber Departmet of Electrical ad Computer Egieerig Uiversity of Wiscosi Madiso Cotets 1 What is a Permutatio 1 2 Cycles 2 2.1 Traspositios 4 3 Orbits

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

THE UNLIKELY UNION OF PARTITIONS AND DIVISORS

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

More information

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

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

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

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

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

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

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

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

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,

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

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

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL. Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

More information

Incremental calculation of weighted mean and variance

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

More information

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

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

More information

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

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

More information

ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE

ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE by Maria E Gageoea ad Silvia Moldoveau

More information

NUMBERS COMMON TO TWO POLYGONAL SEQUENCES

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

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

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

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

Irreducible polynomials with consecutive zero coefficients

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

More information

Distributions of Order Statistics

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

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

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

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

More information

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? 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

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

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

A Recursive Formula for Moments of a Binomial Distribution

A Recursive Formula for Moments of a Binomial Distribution A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,

More information

Lecture 4: Cheeger s Inequality

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

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

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

Ekkehart Schlicht: Economic Surplus and Derived Demand

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/

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

EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND 1 OBTAINED WITH ENGEL SERIES

EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND 1 OBTAINED WITH ENGEL SERIES EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND OBTAINED WITH ENGEL SERIES ELVIA NIDIA GONZÁLEZ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS Abstract. The aciet Egyptias epressed ratioal

More information

THE ABRACADABRA PROBLEM

THE ABRACADABRA PROBLEM 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

More information

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

More information

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

.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,

More information

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

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.;

More information

Sum and Product Rules. Combinatorics. Some Subtler Examples

Sum and Product Rules. Combinatorics. Some Subtler Examples Combiatorics Sum ad Product Rules Problem: How to cout without coutig. How do you figure out how may thigs there are with a certai property without actually eumeratig all of them. Sometimes this requires

More information

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

More information

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:

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

More information

Degree of Approximation of Continuous Functions by (E, q) (C, δ) Means

Degree of Approximation of Continuous Functions by (E, q) (C, δ) Means Ge. Math. Notes, Vol. 11, No. 2, August 2012, pp. 12-19 ISSN 2219-7184; Copyright ICSRS Publicatio, 2012 www.i-csrs.org Available free olie at http://www.gema.i Degree of Approximatio of Cotiuous Fuctios

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

arxiv:1012.1336v2 [cs.cc] 8 Dec 2010

arxiv:1012.1336v2 [cs.cc] 8 Dec 2010 Uary Subset-Sum is i Logspace arxiv:1012.1336v2 [cs.cc] 8 Dec 2010 1 Itroductio Daiel M. Kae December 9, 2010 I this paper we cosider the Uary Subset-Sum problem which is defied as follows: Give itegers

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

Entropy of bi-capacities

Entropy of bi-capacities Etropy of bi-capacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uiv-ates.fr Jea-Luc Marichal Applied Mathematics

More information

5.3. Generalized Permutations and Combinations

5.3. Generalized Permutations and Combinations 53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible

More information

A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY

A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY J. Appl. Prob. 45, 060 070 2008 Prited i Eglad Applied Probability Trust 2008 A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY MARK BROWN, The City College of New York EROL A. PEKÖZ, Bosto Uiversity SHELDON

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

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

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.

More information

THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK

THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E MCCARTHY, SANDRA POTT, AND BRETT D WICK Abstract We provide a ew proof of Volberg s Theorem characterizig thi iterpolatig sequeces as those for

More information

An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process

An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process A example of o-queched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for which

More information

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

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.

More information

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

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

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

THE HEIGHT OF q-binary SEARCH TREES

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

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

1 Introduction to reducing variance in Monte Carlo simulations

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

More information

Multiplexers and Demultiplexers

Multiplexers and Demultiplexers I this lesso, you will lear about: Multiplexers ad Demultiplexers 1. Multiplexers 2. Combiatioal circuit implemetatio with multiplexers 3. Demultiplexers 4. Some examples Multiplexer A Multiplexer (see

More information

Hypothesis testing. Null and alternative hypotheses

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

More information

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

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

More information

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

More information

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.

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

More information

On Formula to Compute Primes. and the n th Prime

On Formula to Compute Primes. and the n th Prime Applied Mathematical cieces, Vol., 0, o., 35-35 O Formula to Compute Primes ad the th Prime Issam Kaddoura Lebaese Iteratioal Uiversity Faculty of Arts ad cieces, Lebao issam.kaddoura@liu.edu.lb amih Abdul-Nabi

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

Figure 40.1. Figure 40.2

Figure 40.1. Figure 40.2 40 Regular Polygos Covex ad Cocave Shapes A plae figure is said to be covex if every lie segmet draw betwee ay two poits iside the figure lies etirely iside the figure. A figure that is ot covex is called

More information

DEFINITION OF INVERSE MATRIX

DEFINITION OF INVERSE MATRIX Lecture. Iverse matrix. To be read to the music of Back To You by Brya dams DEFINITION OF INVERSE TRIX Defiitio. Let is a square matrix. Some matrix B if it exists) is said to be iverse to if B B I where

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

How Euler Did It. Since we discussed the first part of this paper last February, we will start in the middle of this paper, with Euler s Theorem 7:

How Euler Did It. Since we discussed the first part of this paper last February, we will start in the middle of this paper, with Euler s Theorem 7: How Euler Did It Ifiitely may rimes March 2006 Why are there so very may rime umbers? by Ed Sadifer Euclid wodered this more tha 200 years ago, ad his roof that Prime umbers are more tha ay assiged multitude

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

5 Boolean Decision Trees (February 11)

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

More information

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these

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

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

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas: Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

More information

7. Sample Covariance and Correlation

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

More information

Literal Equations and Formulas

Literal Equations and Formulas . Literal Equatios ad Formulas. OBJECTIVE 1. Solve a literal equatio for a specified variable May problems i algebra require the use of formulas for their solutio. Formulas are simply equatios that express

More information