# Bernoulli numbers and the Euler-Maclaurin summation formula

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Physics 6A Witer 006 Beroulli umbers ad the Euler-Maclauri summatio formula I this ote, I shall motivate the origi of the Euler-Maclauri summatio formula. I will also explai why the coefficiets o the right had side of this formula ivolve the Beroulli umbers. First, we defie the Beroulli umbers B. These arise i the Taylor series expasios of x coth(x) ad x cot(x) about x = 0. It is coveiet to defie the ormalizatio of the Beroulli umbers via the Taylor expasio of (x/) coth(x/) as follows: x ( x ) coth = =0 B x, x < π. () ()! This formula oly defies Beroulli umbers with eve o-egative idices. The more commo defiitio is based o the observatio that x ( x ) ] coth = x () e x is a idetity. The, x e x = x B, x < π.! =0 defies all the Beroulli umbers with o-egative idices. Comparig the above formulae, it follows that B = ad B + = 0 for =,, 3,.... For the remaider of this ote, we will oly be cocered with Beroulli umbers of the form B, for o-egative itegers. For the record, we list the first six B here: B 0 =, B = 6, B 4 = 30, B 6 = 4, B 8 = 30, B 0 = 5 66, etc. I geeral, the sigs alterate begiig with B, so that B = ( ) + B, for =,, 3,.... As x 0, both coth(x) ad cot(x) behave as /x; hece we multiply by x i order to have a fuctio with a fiite limit as x 0. To prove this, recall that coth(x) = cosh(x)/ sih(x) where cosh(x) (ex + e x ) ad sih(x) (ex e x ).

3 Here, oe ca either compute the area bouded by the rectagles idicated by the solid lies or by the rectagles idicated by the dashed lies. (I this particular example, = ad = 5.) The former uderestimates the area uder the curve y = f(x), while the latter overestimates this area. That is, I < I < I where I = f( + ) + f( + ) + + f( + ), I = f() + f( + ) + + f( + ). (5) The trapezoidal rule for umerical itegratio taes the average of I ad I. So, we shall mae the approximatio I = (I + I ), which we ca write as: + f(x) dx f() + f( + )] + f( + j). The Euler-Maclauri sum formula arises whe we attempt to covert the above result ito a exact formula. That is, we see to determie a expressio, R, such that: + f(x) dx = f() + f( + )] + j= f( + m) + R. (6) I will show you a sophisticated, yet simple, method for determiig R. You should be forewared that this method is slic ad will gloss over some subtleties that I will metio later. The tric is to itroduce two operators called D ad E. These operators act o a fuctio f(x) ad have very simple defiitios: Df() f (), Ef() f( + ), (7) By act o, I mea that D ad E operate o fuctios. You ca thi of D ad E as little machies. You feed these machies a fuctio ad they will spit out a ew fuctio. 3

4 where as usual, f () (df/dx) x=. With this otatio, eq. (6) reads: + f(x) dx = + E + E + + E + E ]f() + R, sice, e.g., E f() = E Ef() = Ef( + ) = f( + ), etc. Now, for the slic part. We shall write: + E + E + + E + E = + E + E + + E ( E ) = E E ( E ) = (E ) + ], (8) E where we have summed a fiite geometric series i the usual way. But, E is a operator, so what does ( E) mea? The aswer is that we are actually usig a short-had otatio. Whe i doubt, a fuctio of a operator is always defied by its Taylor series. For example, i eq. (8), ( E) = + E + E +. This is a ifiite series, so we should really worry about covergece (what does it mea whe you have a ifiite coverget series of operators rather tha umbers?). For the momet, I will treat these power series expasios as formal objects, ad postpoe questios of covergece util later. So, if you are willig to go alog with this strategy, the we have the followig result: + f(x) dx = (E ) + ] f() + R. (9) E Our ext step is to cosider the Taylor expasio of f(x) about x = : f(x) = f() + f (m) () m! (x ) m. Agai, we should chec for which values of x this series coverges, but we will sidestep this issue agai. If we set x = + i the above expasio, we fid: f( + ) = m=0 f (m) () m! Thus, we ca use our operators D ad E to rewrite this as Ef() = m=0 D m m! f().. 4

5 Note that this formula would be true for ay fuctio f, so we ca coclude that we have a operator idetity: E = m=0 D m m!. (0) I hope you recogize the sum o the right had side of eq. (0). This is the power series expasio of e D. Thus, we coclude that E = e D. () Oce agai, we have itroduced a strage ew object the expoetial of a operator. As before, this is a formal defiitio, ad you should always thi of e D as beig equal to its Taylor series expasio eq. (0)]. The fial step of our aalysis itroduces the idefiite itegral of f(x). Let us call it g(x): g(x) = f(x) dx, or equivaletly f(x) = dg(x) dx. I particular, g () = Dg() = f(). The fudametal theorem of calculus the allows us to write: + f(x) dx = g( + ) g() = (E )g(). Now for the bold move. Sice Dg() = f(), we shall write: This will allow us to write + g() = D f(). f(x) dx = (E ) f(). () D We ow have two differet expressios for + f(x) dx give by eqs. (9) ad (). Sice oly oe of these expressios ivolves R, this meas that we ca ow solve for R. Settig eqs. (9) ad () equal to each other, we obtai: R = (E ) D ] f() E = (E ) D D ( + E )] f(). At this poit, we shall substitute E = e D eq. ()] iside the bracets to obtai R = (E ) ( D D + )] f(). e D 5

6 Usig the idetity give by eq. (), we ca write this last result i a very suggestive way: R = (E ) D D coth D ] f(). Oce agai, we have a fuctio of a operator, which we are istructed to iterpret as a power series. It is at this poit that the Beroulli umbers eter. Usig eq. (), we ca write: D D coth D ] = B m D m (m)!. (3) Notice that at this poit, we oly have o-egative powers of the operator D o the right had side of eq. (3), which we ca easily hadle. Thus, we coclude that: ] R = ( E D m ) B m f(). (m)! We ca write out this expressio more explicitly by usig the defiitios of the operators D ad E eq. (7)]: R = B m f (m ) ( + ) f (m ) () ]. (m)! Isertig this result bac ito eq. (6) yields the followig remarable formula: + f(x) dx = f(+m)+ f()+f(+)] B m f (m ) ( + ) f (m ) () ]. (m)! Notice that this is a exact result. Somehow, we have maaged to tur a formula that started out as a approximatio to a itegral ito a exact result. The fiite sum m f(+m) is also a iterestig object, ad we ca reiterpret the above result as providig a formula for this fiite sum. If we write: f( + m) + f() + f( + )] = f( + m) f() + f( + )], the we ed up with the Euler-Maclauri summatio formula: f( + m) = + f(x) dx + f() + f( + )] + B m f (m ) ( + ) f (m ) () ]. (m)! (4) 6

7 It is ow time to face up to the questio of covergece. The Euler-Maclauri summatio formula as preseted here ivolves a ifiite sum. Give the behavior of the Beroulli umbers B m as m see eq. (4)], it is ot surprisig to lear that i most cases of iterest this is a diverget series. Our derivatio has bee too slic, i that it igored questios of covergece. I fact, oe ca be more careful by replacig all ifiite sums ecoutered above by fiite sums plus remaider terms. By carefully eepig trac of these remaider terms, oe ca obtai a more robust versio of the Euler-Maclauri summatio formula with a remaider term explicitly icluded. This derivatio is beyod the scope of these otes. You ca fid (the more covetioal) derivatio of the Euler-Maclauri summatio formula with remaider term i the textboo by Arfe ad Weber, Mathematical Methods for Physicists. For completeess, I shall display the fial result here: f( + m) = + + f(x) dx + f() + f( + )] p + (p)! B m f (m ) ( + ) f (m ) () ] (m)! 0 B p (x) m=0 f (p) (x + + m) dx, (5) where B p (x) is the Beroulli polyomial of order p (defied o p. 5 of Mc- Quarrie). I may applicatios, the Euler-Maclauri summatio formula provides a asymptotic expasio, i which case the diverget ature of the series i eq. (4) is ot problematical. I other cases, the ifiite sum turs out to be fiite. We shall ed this ote with a few applicatios. For our first example, we tae f(x) = x p ad = 0 i eq. (4). The ifiite sum o the right had side of eq. (4) is i fact fiite i this case, sice f (m ) (x) = 0 for m p +. Evaluatig the derivatives o the right had side of eq. (4), we ca cast the resultig formula ito the followig form: m p = p + p + p/] m=0 ( ) p + B m p+ m, m where p/] is the iteger part of p. For example, if p = the m = = ( + )( + ), 6 6 which reproduces a formula I derived i the first lecture. 7

8 Our secod applicatio is the derivatio of the asymptotic series for l! as. Here, we tae f(x) = l x, with = ad i eq. (4). The itegral i eq. (4) is easily computed: l x dx = l +, ad the derivatives f (m ) (x) are give by: f (m ) (m )! (x) =, m =,, 3,... x Notig that the summatio o the left had side of eq. (4) taes the followig form: we ca write eq. (4) as l( + m) = l + l + + l = l( 3 ) = l! l! = ( + ) l + C + B m m(m ), (6) m where the costat C represets all the remaiig terms of eq. (4) that are idepedet of : B m C = m(m ). (7) Ufortuately, due to the asymptotic behavior of B m as m eq. (4)], the sum i eq. (7) is diverget. However, this is ot surprisig sice we are usig the form of the Euler-Maclauri summatio formula without the remaider term. If we would have icluded the remaider term, the summatios o the right had sides of eqs. (6) ad (7) would have bee fiite sums. I additio, we would have icluded the -idepedet part of the remaider term i the defiitio of C above. I this case, the resultig expressio for C would have bee perfectly well-defied ad fiite. I fact, oe ca aalyze that resultig form for C ad evaluate this costat. However, this requires a umber of additioal trics that lie beyod the scope of these otes. Fially, the -depedet part of the remaider term would appear i eq. (6). By examiig its form c.f. eq. (5)], oe ca prove that the remaider term is of O(/ p ). This meas that eq. (6) is ideed a asymptotic expasio as. Here we shall tae the simpler approach. Namely, we shall simply assume that eq. (6) is a asymptotic expasio as. Comparig this result with Stirlig s approximatio, we coclude that C = l π. Thus, eq. (6) ow reads: l! ( + ) l + l π + 8 B m m(m ),. (8) m

9 This is called Stirlig s asymptotic series. Although the proof give here strictly applies oly for the case of positive iteger, there is a geeralizatio of this derivatio that ca yield the full asymptotic series for l Γ(x + ) for ay real umber x. Not surprisigly, the resultig asymptotic series is idetical to eq. (8) with replaced by x. For our third example, we choose f(x) = /x, with = ad i eq. (4). Agai, the itegral ad (m )-fold derivatives are easily computed: Thus, where dx x = l, f (m ) (m )! (x) =. x m m = l + C + C = + B m m. B m m, (9) m Oce agai, C appears to be diverget. However, as i the previous example, a more careful aalysis icludig the remaider term would produce a fiite expressio for C ad prove that eq. (9) is a asymptotic series. Thus, if we proceed uder this assumptio, we ca compute the costat C by taig the limit of eq. (9): C = lim ( ) m l = γ. (0) We recogize this limit as Euler s costat. Thus, we have derived the followig asymptotic series: m l + γ + B m m m,. By the way, we ca tur this equatio aroud ad use it for a accurate umerical computatio of γ. May other fiite series, summed from m = to, ca be expressed i the form of a asymptotic series as. I will leave it to you as a exercise to wor out the asymptotic series for: m p ζ(p) + p p + + O ( )],, where p > ad ζ(p) is the Riema zeta fuctio. The O(/) term above ca be expressed i terms of a asymptotic series with coefficiets proportioal to the Beroulli umbers, usig the same techiques employed above. 9

10 I our fial example, we choose f(x) = (l x)/x, with = ad. It follows that l m ( ) m l x l l dx + C + x + O C + l + l + O ( ) l, () where C represets the -idepedet pieces of eq. (4), ad the O(l / ) remaider correspods to a sum with Beroulli umber coefficiets (which we do ot write out explicitly here). Thus, C = lim ( l m m l This result ca be used to evaluate the sum S ( ) l. = ). () Cosider the sum S N made up of the first N terms of the above series. We ca write: = S N = N = = ( ) l N = l + N = N = = l l + N + l N = = l, (3) where we have used l() = l + l i obtaiig the secod lie of eq. (3). However, eqs. () ad () ad eqs. (9) ad (0) imply respectively that: N ( ) l N ( ) = C + l N + O, N = γ + l N + O. N Isertig these result ito eq. (3), the costat C drops out, ad we fid: ( ) S N = l (N) + l N + l (γ + l N) + O N = l ( ( ) γ l ) + O. N Taig the limit of N, we obtai the desired sum S = lim N S N : ( ) l = l ( γ l ). = This result (the = term does ot cotribute to the sum) was previously oted i the Riema zeta fuctio hadout. 0 =

### MAT1026 Calculus II Basic Convergence Tests for Series

MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real

### INFINITE SEQUENCES AND SERIES

11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS

### 62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

### Sequences, Series, and All That

Chapter Te Sequeces, Series, ad All That. Itroductio Suppose we wat to compute a approximatio of the umber e by usig the Taylor polyomial p for f ( x) = e x at a =. This polyomial is easily see to be 3

### Solutions to Final Exam Review Problems

. Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the

### Math 2784 (or 2794W) University of Connecticut

ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS

MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak

### Series III. Chapter Alternating Series

Chapter 9 Series III With the exceptio of the Null Sequece Test, all the tests for series covergece ad divergece that we have cosidered so far have dealt oly with series of oegative terms. Series with

### MA131 - Analysis 1. Workbook 9 Series III

MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................

### AP Calculus Chapter 9: Infinite Series

AP Calculus Chapter 9: Ifiite Series 9. Sequeces a, a 2, a 3, a 4, a 5,... Sequece: A fuctio whose domai is the set of positive itegers = 2 3 4 a = a a 2 a 3 a 4 terms of the sequece Begi with the patter

### 2.4.2 A Theorem About Absolutely Convergent Series

0 Versio of August 27, 200 CHAPTER 2. INFINITE SERIES Add these two series: + 3 2 + 5 + 7 4 + 9 + 6 +... = 3 l 2. (2.20) 2 Sice the reciprocal of each iteger occurs exactly oce i the last series, we would

### 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

### Bernoulli Numbers. n(n+1) = n(n+1)(2n+1) = n(n 1) 2

Beroulli Numbers Beroulli umbers are amed after the great Swiss mathematiia Jaob Beroulli5-705 who used these umbers i the power-sum problem. The power-sum problem is to fid a formula for the sum of the

### (a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b)

Chapter 0 Review 597. E; a ( + )( + ) + + S S + S + + + + + + S lim + l. D; a diverges by the Itegral l k Test sice d lim [(l ) ], so k l ( ) does ot coverge absolutely. But it coverges by the Alteratig

### Section 11.8: Power Series

Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i

### Integrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number

MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios

### sin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =

60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece

### MATH 10550, EXAM 3 SOLUTIONS

MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,

### De la Vallée Poussin Summability, the Combinatorial Sum 2n 1

J o u r a l of Mathematics ad Applicatios JMA No 40, pp 5-20 (2017 De la Vallée Poussi Summability, the Combiatorial Sum 1 ( 2 ad the de la Vallée Poussi Meas Expasio Ziad S. Ali Abstract: I this paper

### A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as

### Sequences of Definite Integrals, Factorials and Double Factorials

47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology

### MA131 - Analysis 1. Workbook 2 Sequences I

MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................

### AP Calculus. Notes and Homework for Chapter 9

AP Calculus Notes ad Homework for Chapter 9 (Mr. Surowski) I do t feel that the textbook does a particularly good job at itroducig the material o ifiite series, power series (Maclauri ad Taylor series),

### Dirichlet s Theorem on Arithmetic Progressions

Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty

### Math 106 Fall 2014 Exam 3.2 December 10, 2014

Math 06 Fall 04 Exam 3 December 0, 04 Determie if the series is coverget or diverget by makig a compariso (DCT or LCT) with a suitable b Fill i the blaks with your aswer For Coverget or Diverget write

### Sequences I. Chapter Introduction

Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which

### The Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1

460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that

### Solutions to Tutorial 5 (Week 6)

The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial 5 (Wee 6 MATH2962: Real ad Complex Aalysis (Advaced Semester, 207 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/

### Zeros of Polynomials

Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

### Singular Continuous Measures by Michael Pejic 5/14/10

Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable

### Section 11.6 Absolute and Conditional Convergence, Root and Ratio Tests

Sectio.6 Absolute ad Coditioal Covergece, Root ad Ratio Tests I this chapter we have see several examples of covergece tests that oly apply to series whose terms are oegative. I this sectio, we will lear

### Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series

### Fourier Series and the Wave Equation

Fourier Series ad the Wave Equatio We start with the oe-dimesioal wave equatio u u =, x u(, t) = u(, t) =, ux (,) = f( x), u ( x,) = This represets a vibratig strig, where u is the displacemet of the strig

### Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. Stirling s Formula Derived from the Gamma Function

Steve R. Dubar Departmet of Mathematics 23 Avery Hall Uiversity of Nebraska-Licol Licol, NE 68588-3 http://www.math.ul.edu Voice: 42-472-373 Fax: 42-472-8466 Topics i Probability Theory ad Stochastic Processes

### Analytic Continuation

Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for

### 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,

### Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.

Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio

### a 3, a 4, ... are the terms of the sequence. The number a n is the nth term of the sequence, and the entire sequence is denoted by a n

60_090.qxd //0 : PM Page 59 59 CHAPTER 9 Ifiite Series Sectio 9. EXPLORATION Fidig Patters Describe a patter for each of the followig sequeces. The use your descriptio to write a formula for the th term

### Quiz No. 1. ln n n. 1. Define: an infinite sequence A function whose domain is N 2. Define: a convergent sequence A sequence that has a limit

Quiz No.. Defie: a ifiite sequece A fuctio whose domai is N 2. Defie: a coverget sequece A sequece that has a limit 3. Is this sequece coverget? Why or why ot? l Yes, it is coverget sice L=0 by LHR. INFINITE

### w (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.

2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For

### PHY4905: Nearly-Free Electron Model (NFE)

PHY4905: Nearly-Free Electro Model (NFE) D. L. Maslov Departmet of Physics, Uiversity of Florida (Dated: Jauary 12, 2011) 1 I. REMINDER: QUANTUM MECHANICAL PERTURBATION THEORY A. No-degeerate eigestates

### Analysis of Algorithms. Introduction. Contents

Itroductio The focus of this module is mathematical aspects of algorithms. Our mai focus is aalysis of algorithms, which meas evaluatig efficiecy of algorithms by aalytical ad mathematical methods. We

### Q-BINOMIALS AND THE GREATEST COMMON DIVISOR. Keith R. Slavin 8474 SW Chevy Place, Beaverton, Oregon 97008, USA.

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 Q-BINOMIALS AND THE GREATEST COMMON DIVISOR Keith R. Slavi 8474 SW Chevy Place, Beaverto, Orego 97008, USA slavi@dsl-oly.et Received:

### SUMMARY OF SEQUENCES AND SERIES

SUMMARY OF SEQUENCES AND SERIES Importat Defiitios, Results ad Theorems for Sequeces ad Series Defiitio. A sequece {a } has a limit L ad we write lim a = L if for every ɛ > 0, there is a correspodig iteger

### Math 106 Fall 2014 Exam 3.1 December 10, 2014

Math 06 Fall 0 Exam 3 December 0, 0 Determie if the series is coverget or diverget by makig a compariso DCT or LCT) with a suitable b Fill i the blaks with your aswer For Coverget or Diverget write Coverget

### Practice Test Problems for Test IV, with Solutions

Practice Test Problems for Test IV, with Solutios Dr. Holmes May, 2008 The exam will cover sectios 8.2 (revisited) to 8.8. The Taylor remaider formula from 8.9 will ot be o this test. The fact that sums,

### Read carefully the instructions on the answer book and make sure that the particulars required are entered on each answer book.

THE UNIVERSITY OF WARWICK FIRST YEAR EXAMINATION: Jauary 2009 Aalysis I Time Allowed:.5 hours Read carefully the istructios o the aswer book ad make sure that the particulars required are etered o each

### Regression with an Evaporating Logarithmic Trend

Regressio with a Evaporatig Logarithmic Tred Peter C. B. Phillips Cowles Foudatio, Yale Uiversity, Uiversity of Aucklad & Uiversity of York ad Yixiao Su Departmet of Ecoomics Yale Uiversity October 5,

### ECE 901 Lecture 12: Complexity Regularization and the Squared Loss

ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality

### Physics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.

Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;

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

### If we want to add up the area of four rectangles, we could find the area of each rectangle and then write this sum symbolically as:

Sigma Notatio: If we wat to add up the area of four rectagles, we could fid the area of each rectagle ad the write this sum symbolically as: Sum A A A A Liewise, the sum of the areas of te triagles could

### Once we have a sequence of numbers, the next thing to do is to sum them up. Given a sequence (a n ) n=1

. Ifiite Series Oce we have a sequece of umbers, the ext thig to do is to sum them up. Give a sequece a be a sequece: ca we give a sesible meaig to the followig expressio? a = a a a a While summig ifiitely

### Math 21B-B - Homework Set 2

Math B-B - Homework Set Sectio 5.:. a) lim P k= c k c k ) x k, where P is a partitio of [, 5. x x ) dx b) lim P k= 4 ck x k, where P is a partitio of [,. 4 x dx c) lim P k= ta c k ) x k, where P is a partitio

### Section 1 of Unit 03 (Pure Mathematics 3) Algebra

Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course

### Complex Analysis Spring 2001 Homework I Solution

Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle

### (b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?

MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle

### Frequency Response of FIR Filters

EEL335: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we itroduce the idea of the frequecy respose of LTI systems, ad focus specifically o the frequecy respose of FIR filters.. Steady-state

### Lecture 3 The Lebesgue Integral

Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified

EECS 598: Statistical Learig Theory, Witer 204 Topic 0 Rademacher Complexity Lecturer: Clayto Scott Scribe: Ya Deg, Kevi Moo Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved for

### 6. Uniform distribution mod 1

6. Uiform distributio mod 1 6.1 Uiform distributio ad Weyl s criterio Let x be a seuece of real umbers. We may decompose x as the sum of its iteger part [x ] = sup{m Z m x } (i.e. the largest iteger which

### Recurrence Relations

Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The

### SEQUENCES AND SERIES

9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first

### 11.6 Absolute Convergence and the Ratio and Root Tests

.6 Absolute Covergece ad the Ratio ad Root Tests The most commo way to test for covergece is to igore ay positive or egative sigs i a series, ad simply test the correspodig series of positive terms. Does

### Lesson 10: Limits and Continuity

www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

### Math 4400/6400 Homework #7 solutions

MATH 4400 problems. Math 4400/6400 Homewor #7 solutios 1. Let p be a prime umber. Show that the order of 1 + p modulo p 2 is exactly p. Hit: Expad (1 + p) p by the biomial theorem, ad recall from MATH

### In this section, we show how to use the integral test to decide whether a series

Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Itegral Test I this sectio, we show how to use the itegral test to decide

### A PROOF OF THE TWIN PRIME CONJECTURE AND OTHER POSSIBLE APPLICATIONS

A PROOF OF THE TWI PRIME COJECTURE AD OTHER POSSIBLE APPLICATIOS by PAUL S. BRUCKMA 38 Frot Street, #3 aaimo, BC V9R B8 (Caada) e-mail : pbruckma@hotmail.com ABSTRACT : A elemetary proof of the Twi Prime

### Solutions to Math 347 Practice Problems for the final

Solutios to Math 347 Practice Problems for the fial 1) True or False: a) There exist itegers x,y such that 50x + 76y = 6. True: the gcd of 50 ad 76 is, ad 6 is a multiple of. b) The ifiimum of a set is

### (A) 0 (B) (C) (D) (E) 2.703

Class Questios 007 BC Calculus Istitute Questios for 007 BC Calculus Istitutes CALCULATOR. How may zeros does the fuctio f ( x) si ( l ( x) ) Explai how you kow. = have i the iterval (0,]? LIMITS. 00 Released

### Topic 9 - Taylor and MacLaurin Series

Topic 9 - Taylor ad MacLauri Series A. Taylors Theorem. The use o power series is very commo i uctioal aalysis i act may useul ad commoly used uctios ca be writte as a power series ad this remarkable result

### Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence

Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i

### BINOMIAL COEFFICIENT AND THE GAUSSIAN

BINOMIAL COEFFICIENT AND THE GAUSSIAN The biomial coefficiet is defied as-! k!(! ad ca be writte out i the form of a Pascal Triagle startig at the zeroth row with elemet 0,0) ad followed by the two umbers,

### Bernoulli, Ramanujan, Toeplitz e le matrici triangolari

Due Giori di Algebra Lieare Numerica www.dima.uige.it/ dibeede/gg/home.html Geova, 6 7 Febbraio Beroulli, Ramauja, Toeplitz e le matrici triagolari Carmie Di Fiore, Fracesco Tudisco, Paolo Zellii Speaker:

### Monte Carlo Integration

Mote Carlo Itegratio I these otes we first review basic umerical itegratio methods (usig Riema approximatio ad the trapezoidal rule) ad their limitatios for evaluatig multidimesioal itegrals. Next we itroduce

### The standard deviation of the mean

Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider

### Sequences III. Chapter Roots

Chapter 4 Sequeces III 4. Roots We ca use the results we ve established i the last workbook to fid some iterestig limits for sequeces ivolvig roots. We will eed more techical expertise ad low cuig tha

### Seunghee Ye Ma 8: Week 5 Oct 28

Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value

### Definition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.

4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad

### arxiv: v2 [math.nt] 10 May 2014

FUNCTIONAL EQUATIONS RELATED TO THE DIRICHLET LAMBDA AND BETA FUNCTIONS JEONWON KIM arxiv:4045467v mathnt] 0 May 04 Abstract We give closed-form expressios for the Dirichlet beta fuctio at eve positive

### The Gamma function Michael Taylor. Abstract. This material is excerpted from 18 and Appendix J of [T].

The Gamma fuctio Michael Taylor Abstract. This material is excerpted from 8 ad Appedix J of [T]. The Gamma fuctio has bee previewed i 5.7 5.8, arisig i the computatio of a atural Laplace trasform: 8. ft

### P1 Chapter 8 :: Binomial Expansion

P Chapter 8 :: Biomial Expasio jfrost@tiffi.kigsto.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 6 th August 7 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework

### 0.1. Geometric Series Formula. This is in your book, but I thought it might be helpful to include here. If you have a geometric series

Covergece tests These otes discuss a umer of tests for determiig whether a series coverges or 0.. Geometric Series Formula. This is i your oo, ut I thought it might e helpful to iclude here. If you have

### INEQUALITIES BJORN POONEN

INEQUALITIES BJORN POONEN 1 The AM-GM iequality The most basic arithmetic mea-geometric mea (AM-GM) iequality states simply that if x ad y are oegative real umbers, the (x + y)/2 xy, with equality if ad

### HOMEWORK #10 SOLUTIONS

Math 33 - Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous

### REVIEW 1, MATH n=1 is convergent. (b) Determine whether a n is convergent.

REVIEW, MATH 00. Let a = +. a) Determie whether the sequece a ) is coverget. b) Determie whether a is coverget.. Determie whether the series is coverget or diverget. If it is coverget, fid its sum. a)

### 11. FINITE FIELDS. Example 1: The following tables define addition and multiplication for a field of order 4.

11. FINITE FIELDS 11.1. A Field With 4 Elemets Probably the oly fiite fields which you ll kow about at this stage are the fields of itegers modulo a prime p, deoted by Z p. But there are others. Now although

### A MODIFIED BERNOULLI NUMBER. D. Zagier

A MODIFIED BERNOULLI NUMBER D. Zagier The classical Beroulli umbers B, defied by the geeratig fuctio x e x = = B x!, ( have may famous ad beautiful properties, icludig the followig three: (i B = for odd

### MAS111 Convergence and Continuity

MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece

### Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t

Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said

### BC: Q401.CH9A Convergent and Divergent Series (LESSON 1)

BC: Q40.CH9A Coverget ad Diverget Series (LESSON ) INTRODUCTION Sequece Notatio: a, a 3, a,, a, Defiitio: A sequece is a fuctio f whose domai is the set of positive itegers. Defiitio: A ifiite series (or

### EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS

EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS TUTORIAL 1 - DIFFERENTIATION Use the elemetary rules of calculus arithmetic to solve problems that ivolve differetiatio

### The Arakawa-Kaneko Zeta Function

The Arakawa-Kaeko Zeta Fuctio Marc-Atoie Coppo ad Berard Cadelpergher Nice Sophia Atipolis Uiversity Laboratoire Jea Alexadre Dieudoé Parc Valrose F-0608 Nice Cedex 2 FRANCE Marc-Atoie.COPPO@uice.fr Berard.CANDELPERGHER@uice.fr

### ANOTHER GENERALIZED FIBONACCI SEQUENCE 1. INTRODUCTION

ANOTHER GENERALIZED FIBONACCI SEQUENCE MARCELLUS E. WADDILL A N D LOUIS SACKS Wake Forest College, Wisto Salem, N. C., ad Uiversity of ittsburgh, ittsburgh, a. 1. INTRODUCTION Recet issues of umerous periodicals

### Probability, Expectation Value and Uncertainty

Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such

### In number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.

Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers

### ECE 901 Lecture 4: Estimation of Lipschitz smooth functions

ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig

### 17 Phonons and conduction electrons in solids (Hiroshi Matsuoka)

7 Phoos ad coductio electros i solids Hiroshi Matsuoa I this chapter we will discuss a miimal microscopic model for phoos i a solid ad a miimal microscopic model for coductio electros i a simple metal.