Mathematical Methods for Physics and Engineering

Mathematical Methods for Physics ad Egieerig Lecture otes Sergei V. Shabaov Departmet of Mathematics, Uiversity of Florida, Gaiesville, FL 326 USA

CHAPTER The theory of covergece. Numerical sequeces.. Defiitio of a coverget sequece. A ifiite collectio of real or complex umbers {z }, where a positive iteger eumerates the elemets, is called a sequece. Oe ca also say that a sequece is a real or complex valued fuctio o a set of positive itegers, that is, it is a rule that assigs a uique umber z to every positive iteger. Defiitio.. If there exists a umber L such that for ay positive umber ε > 0 oe ca fid a positive iteger N such that z L < ε, > N, the the sequece {z } is said to coverge ad the umber L is called the it of the sequece. I this case, oe writes z = L or z L as The defiitio implies that: there are fiitely may terms of the sequece outside ay eighborhood of the it L; ay coverget sequece is bouded: z = L z M, for some umber M idepedet of ; if the it of a sequece exists, the it is uique. The reader is advised to prove the above assertios..2. Basic coverget sequeces. It ca be proved that for ay positive p > 0 ad ay real α = p = p = 0 α + p) = 0 3

4. THE THEORY OF CONVERGENCE Covergece to ifiity. If for ay positive umber M > 0, oe ca fid a iteger N such that z > M, > N the the sequece {z } is said to ted to ifiity as, ad oe writes z = or z as. Let {x } be a sequece of real umbers. If for ay positive umber M oe ca fid a iteger N such that x < M for all > N, the the sequece is said to ted to egative ifiity, ad oe writes x = or x as..3. Order of magitude. If two sequeces {z } ad {ξ } are such that ξ < M, > N z for some umber M idepedet of ad some iteger N, the the sequece are said to be of the same order of magitude ad oe writes ξ = Oz ). If ξ /z 0 as, the oe writes For example, 5 + 3 4 + 2 + = O ξ = oz ) ), 3.4. Basic it laws. Suppose that The z = A ad ) 3 2 2 + = o. ξ = B z + ξ ) = z + ξ = A + B z ξ ) = z ) ξ ) = AB z = z = A ξ ξ B ; i the latter equality it is assumed, i additio, that B 0 ad ξ 0. Note that the coverse is false. The reader is advised to give examples whe the its of the sum or product or ratio of two sequeces exists, but the its of each sequeces do ot.

. NUMERICAL SEQUENCES 5.5. Upper ad lower its. Let E be a set of real umbers. A set E is bouded from above if there is a umber b such that x b, x E, ad the umber b is called a upper boud of E. A set E is bouded from below if there is a umber a such that a x, x E, ad the umber a is called a lower boud of E. A set E is said to be bouded if it has lower ad upper bouds. Defiitio.2. supremum ad ifimum of a set of real umbers) Let E be a bouded set. The the least upper boud ad the greatest lower boud of E are called, respectively, the supremum ad ifimum of E ad deoted as sup E ad if E. If E is ot bouded from above, the supe = ad, if E is ot bouded from below, the if E =. I other words, supe is a upper boud of E, while sup E ɛ is ot a upper boud for ay ɛ > 0. Similarly, if E is a lower boud of E, but if E + ɛ is ot a lower boud of E for ay ɛ > 0. The umbers sup E ad if E are uique. They may or may ot be i E. For example, if E = [0, ], the sup E = ad if E = 0 are i E, while for E = 0, ) a ope iterval), sup E = ad if E = 0 are ot i E. Defiitio.3. a it poit of a sequece) A umber x is a it poit of real sequece {x } if ay eighborhood of x cotais ifiitely may terms of {x }: for ifiitely may s. x x < ε, ε > 0, For example, the sequece x = ) has two it poits, ±. Defiitio.4. upper ad lower its of a sequece) Let E be the set of all it poits of a real sequece {x }. The sup E ad if E are called the upper ad lower its of the sequece, respectively, ad deoted as sup E = supx, For example, sup ) + ) if E = supx =, sup ) + ) =. A real sequece coverges if ad oly if its upper ad lower its exists ad coicide. Theorem.. Bolzao) A bouded real sequece has at least oe it poit.

6. THE THEORY OF CONVERGENCE.6. Icreasig sequeces. A sequece of real umbers {x } is called a icreasig sequece if x + x. Theorem.2. A icreasig sequece either coverges or ted to ifiity. I particular, if x < M for some M idepedet o, the {x } coverges, that is, ay bouded icreasig sequece coverges. Ideed, if x < M, the there exists sup{x } = L. Sice x is a icreasig sequece, oly fiitely may terms of it are ot i a iterval L ε, L]. This implies that x = L. If {x } is ot bouded from above, the = sup{x } =. 2. Series For a real or complex sequece {z }, a formal ifiite sum z = z + z 2 + z 3 + is called a series, ad a sequece {S } : S = z k = z + z 2 + z. k= is called a sequece of partial sums of the series z. Defiitio 2.. the sum of a series) A series z is said to coverge to a umber S if the sequece of partial sums coverges to S, ad, i this case, the umber S is called the sum of the series: z = S = S. If the sequece of partial sums has o it, the series is said to diverge. 2.. Necessary coditio for covergece. Suppose that a series z coverges. The z 0 as. Ideed, let z = S = S The by the Cauchy criterio for covergece of a sequece, for ay ε > 0 oe ca fid a iteger N such that z + = S + S < ε, > N z = 0.

2. SERIES 7 For example, the followig series diverge: q + 3 2 2 + ) q, for ay positive itegers p ad q. Explai why! q log ) p 2.2. Geometric series. Let us ivestigate the covergece of the complex series z = + z + z 2 + z 3 +, =0 where z is a complex umber. It is called a geometric series. Clearly, the series diverges if z =. If z, by the idetity z = z) + z + + z ) S = + z + z 2 + + z = z z the sequece of partial sums {S } coverges =0 z z = z =, z <, z because z 0 as if ad oly if z <. 2.3. Absolute ad coditioal covergece. Let z be coverget. If the series of absolute values z coverges, the the series z is called absolutely coverget, otherwise z is said to coverge coditioally. The geometric series coverges absolutely for all z <, while for z =, it coverges coditioally. 2.4. Series of o-egative terms. If a 0 for all, the the series a is called a series of o-egative terms. Theorem 2.. Necessary ad sufficiet coditio for covergece) Let 0 a + a for all the sequece {a } is mootoically decreasig). The the series a coverges if ad oly if the series 2 k a 2 k coverges: a < 2 k a 2 k <. k=0

8. THE THEORY OF CONVERGENCE By this criterio, the followig series coverge <, p >, p <, p >, log ) p =2 log)loglog)) <, For example, =3 =3 log)loglog))) 2 <. 2 k 2 k ) = ) k 2 p < p k=0 k=0 if ad oly if 2 p < or p < 0 or p >. Therefore the series / p coverges for p > ad diverges for p. 2.5. Compariso test. If z x for all > N for some N ad the series x coverges, the the complex series z coverges, too. If u x 0 ad the series x diverges, the the real series u diverges, too. Ideed, cosider the sequece of partial sums for z : S = z + z 2 + + z z + z 2 + + z x + x 2 + + x x = A. The sequece { k= z k } is bouded by A ad icreasig mootoically. Therefore it coverges, which meas that the series z coverges absolutely ad the absolute covergece implies covergece). For example, the series cosx) p coverges absolutely for all real x ad p > because cosx) p ad p <. p

2. SERIES 9 The series =0 z ) 2 si 3 coverges absolutely for all complex z. Ideed, sice siu)/u as u 0, z ) z ) 3 si = z 3 si M, 3 3 for some M idepedet of because every coverget sequece is bouded. Therefore z ) ) 2 2 3 z ) si = 2 si M 3 3 3 3) ad =0 2/3) = 3 < as the sum of a geometric series. 2.6. Root ad ratio tests. Theorem 2.2. Root test) Let The α = sup z. i) α < z coverges absolutely ii) α > z diverges iii) α = the test gives o iformatio For example, the geometric series z coverges for all z < by the root test. However, the root test gives o iformatio about the series / p because as. Theorem 2.3. Ratio test) sup z + z < z coverges absolutely z + z, > N z diverges z + z = the test gives o iformatio

0. THE THEORY OF CONVERGENCE The ratio test is coveiet to use if terms of the series cotai factorials. For example, the series 2 z coverges absolutely for all complex z. Ideed, z + + )2! z = 2 + )! z = + z 0 as 2 for ay z. The use of the root test would require to study the asymptotic behavior of!. This ca be doe with the help of Stirlig s formula: e! 2π =. It is ot difficult to show usig Stirlig s formula that For the above series z =!! = e. 2 z = z! = z!! = z e = 0. 2.7. Wider scope of the root test. The ratio test is easier to apply tha the root test. However, the root test has a wider scope. Let b > a >. Cosider the series u = a + b + a + 2 b + 2 a + 3 b + 3 A geeral term of the series ca be writte as { a u = k, = 2k b k, k =, 2,..., = 2k Oe has if u + u a ) k = = 0 k b ) k b = a u + sup = u k So, the ratio test does ot eve apply. However, the root test idicates covergece: sup 2k u = k a = <. k a

2. SERIES The followig results make a descriptio of a wider scope of the root test more precise. Theorem 2.4. Wider scope of the root test) Let {x } be a sequece of positive umbers, c > 0. The sup if x + x sup x x + x if x The first iequality shows that the root test may idicate covergece whe the ratio test does ot do so. 2.8. De Morga test. The ratio test gives o iformatio if z + / z as. It turs out that i this case the covergece depeds o how fast the ratio approaches. Theorem 2.5. De Morga test) Suppose that sup z + =, z ) z+ = c, for some c > 0, z The the series z coverges absolutely. Corollary 2.. If z + z = + A ) + O, A < 2 the the series z coverges absolutely. For example, the series p coverges for p >. However, the ratio or root test fails to detect covergece. The De Morga test shows covergece: p + ) p = + )p = p + O ) 2 ad, hece, the series coverges if A = p < or p >. The reader is advised to ivestigate the series ) r exp q k whe r > q ad r < q. k=

2. THE THEORY OF CONVERGENCE 2.9. Coditioally coverget series. Theorem 2.6. Abel test) Let the sequece {A }, A = k= a k, be bouded, ad the sequece {b } is decreasig mootoically, b b 2 b 3, so that b = 0. The the series a b coverges. By the root test, the series coverges absolutely i the disk z < i the complex plae. Let us ivestigate the covergece o the boudary of the disk. Put z = e iθ so that a k = e ikθ ad b = / 0 mootoically as. The z A = e iθ + e 2iθ + + e iθ iθ eiθ = e e iθ A = e iθ e iθ cosθ) = cos θ cosθ) siθ/2) Thus, the sequece {A } is bouded for all 0 < θ < 2π, ad the series i questio coverges at all poits of the closed disk z except z =. The covergece o the boudary is ot absolute. Corollary 2.2. Alteratig series) Suppose that c is decreasig mootoically, c + c, so that c 0 as, ad c 2m 0, c 2m 0. The the series c coverges. This is a simple cosequece of the previous theorem. Ideed, put a = ) ad b = c. The A ad b = / 0 mootoically. By the alteratig series test, the series ) + coverges, but ot absolutely. 2.0. Rearragemets. Cosider a oe-to-oe map of the set of all positive iteger oto itself so that {, 2, 3,...} is mapped to {k, k 2, k 3,...}. The series a k = a k + a k2 + a k3 + = a + a 2 + a 3 + = a is called a rearragemet of the series a = a + a 2 + a 3 +.

2. SERIES 3 Theorem 2.7. Rearragemet ad absolute covergece) A rearragemet of ay absolutely coverget series coverges absolutely to the same sum: a = a. I cotrast, a rearragemet of a coditioally coverget series ca coverge to a differet sum ad eve fail to coverge. For example, cosider the alteratig series ) + = 2 + 3 4 + 5 6 + = S ad let S be the sequece of its partial sums so that S S as. Evidetly, S < S 3 = 2 + 3 = 5 6. Cosider the rearragemet of the series i which two positive terms are followed by oe egative: a = + 3 2 + 5 + 7 6 + ad let S be the sequece of its partial sums. It follows that 4k 3 + 4k 2k > 0 S 3 = S 3 < S 6 < S 9 < Therefore there are ifiitely may terms of the sequece S greater tha 5/6. This implies that sups > S 3 = 5 6 > S ad, hece, the sequece S caot coverge to S. Theorem 2.8. Riema) Let a coverges but ot absolutely. For ay two elemets of the exteded real umber system α β, there exists a rearragemet a with partial sums S such that if S = α, sups = β The Riema theorem about rearragemets shows that a rearragemet of a coditioally coverget series may be costructed to coverge to ay desired umber α = β, or diverge to ±, or to have a sequece of partial sums that does ot coverge ad oscillates betwee

4. THE THEORY OF CONVERGENCE ay two umbers α < β) or has ubouded oscillatios, α =, β =, or both). Such a behavior of coditioally coverget series stems from the followig property. Theorem 2.9. Properties of coditioally coverget series) Let a be a real series that coverges but ot absolutely. If S p + is the sum of the first p positive terms of the series ad Sq is the sum of its first q egative terms, the p S+ p =, q S q =. For example, ) + = 2 + 3 4 + 5 6 + p S+ p = + p 3 + + ) = 2p p= q S q = q 2 4 ) = 2q 2 2p = q= q = Note that a real coditioally coverget series ecessarily cotais ifiitely may egative terms ad ifiitely may positive terms otherwise it were a absolutely coverget series). As the series made of positive ad egative terms of a coditioally coverget series diverge to ifiity, the sum of the series looks like a udetermied form whose value depeds o how the ifiities i this form are regularized, or, more precisely, how egative ad positive terms are ordered whe computig the sequece of partial sums. For example, let us costruct a rearragemet that coverges to a give umber β > 0 the case β 0 ca be cosidered alog a similar lie of argumets). Let S deote the sequece of partial sums of the sought-after rearragemet. The rearragemet should be such that S β as. Let {a + } ad {a } deote the subsequeces of positive ad egative terms i the sequece {a }. Clearly, these sequeces ca be reordered so that they coverge to 0 mootoically. Note that a 0 as owig to the coditioal covergece of a. Let us keep the same otatios for the reordered mootoic) sequeces, a ± + a ± for all. Put a = a + for =, 2,..., m + where m is determied by the coditio S < β m + S. It is m + always possible to fulfill this coditio for ay β > 0 because a + diverges. The put a = m + + a for =, 2,..., m where m is

2. SERIES 5 determied by the coditio S < β S m +m+ m +m+. The iteger m always exists as a =. Next, positive terms a+, > m+, are added util the value of S exceeds β, after that the egative terms a, > m, are added util the value of S drops below β ad so o to obtai a sequece S oscillatig about β: {a } = {a +,..., a +, a m +,..., a, a + m m + +,..., a+, a m + +m+ 2 m +,..., a,...}. m +m 2 The sequece of partial sums of a oscillates about β with the amplitude of oscillatios decreasig to zero, which implies that S β as. Ideed, the procedure geerates two sequeces of itegers m ± k, k =, 2,..., which determie the rearragemet so that m ± = m ± + m ± 2 + + m ± k, m = m+ + m, S < β m m k S or 0 S β < a + m m k m m m k + S m < β S m or 0 < β S m a m. Sice a ± 0 mootoically as, it is cocluded that the values of S overshoot β by o more tha a+ ad udershoot β by o less m + tha a for all > m. I other words, the amplitude of oscillatios m of the sequece S is mootoically decreasig to 0 with icreasig the umber of steps k because m ± as k so that a ± m 0 ± ad, hece, S β as. The reader is advised to carry out this procedure explicitly for the alteratig harmoic series ) + /. 2.. Double series. Cosider a ifiite table of umbers real or complex) u u 2 u 3 u 2 u 22 u 23 u 3 u 32 u 33.. The collectio of elemets u m is called a double sequece. The formal sum of all elemets of the sequece {u m } is called a double series ad deoted as u m. m, Defiitio 2.2. the sum of a double series) A double series is said to coverge to a umber S if for ay ε > 0 oe.

6. THE THEORY OF CONVERGENCE ca fid two itegers N ad M such that for all > N ad m > M m S S m < ε, S m = u ij. i= j= The umber S is called the sum of the double series. The sequece {S m } is also called a sequece of partial sums. A double series is said to coverge absolutely if the series of absolute values coverges,,m u m <. Theorem 2.0. Stolz) A double series coverges if ad oly if for ay ε > 0 there exist two itegers N ad M such that for all, m. S M+m,N+ S MN < ε Stolz theorem is othig but the aalog of the Cauchy criterio for sequeces. Its advatage is that the covergece ca be established without ay kowledge about the value of the sum. Clearly, the terms of a coverget double series must ted to zero as ad m. This comprises the ecessary coditio for covergece: u m = S u m = 0 m,) Put m, S m = S = u m, S = u m, m= S m, m= S = S If S ad S exist, the they are called the sum of the double series by rows ad by colums, respectively. I geeral, ) ) S S or u m u m For example, if m= m= u m = m m + )!,, m > 0; 2 m+ m!! u m0 = 2 m, u 0 = 2, u 00 = 0,

2. SERIES 7 The ) u m =, m=0 =0 ) u m =. =0 m=0 Theorem 2.. Prigsheim) If a double series coverges to S ad the sums by rows ad colums exist, the they are equal to S. Theorem 2.2. Cauchy) Suppose that series a = A ad b m = B coverge absolutely. The the double series a b m = a b + a 2 b + a b 2 +, cosistig of the products a b m writte i ay order, coverges absolutely ad its sum is equal to AB. For example, if a = + z 2 + z2 2 + z3 2 2 +, b = + 3 z + z + 2 z + 3 The the series a b m represeted as a series over powers of z coverges absolutely for all z 2. 2.2. Power series. Let {a } 0 series is called a power series. Put be a sequece of complex umbers. The a z =0 R = α, α = sup a The upper it may take values i the exteded system of real umbers. If α = 0, the R = ad, if α =, the R = 0. The quatity R is called the radius of covergece of the power series. This term is justified by the followig property of R. Theorem 2.3. A power series a z coverges absolutely if z < R ad diverges if z > R. Ideed, by the root test sup a z = z sup a = z α = z R the power series coverges if z /R < or z < R ad diverges if z > R. I the complex plae, a power series coverges i the iterior

8. THE THEORY OF CONVERGENCE of the disk of radius R cetered at the origi. For the boudary poits z = R, the series may coverge or diverge. The radius of coverges of the series z is R = sup ) = ) =. O the boudary of the uit disk z = e iθ, 0 θ 2π, the series was show above to coverge coditioally for all 0 < θ < 2π for all poits of the boudary except z = ) 2.3. Ifiite products. Let a sequece { + a } have o zero terms. Cosider a sequece of products p = + a ) + a 2 ) + a ), =, 2,... If the it p = p exists, the the umber p is called the value of the ifiite product + a ) + a 2 ) + a 3 ) = + a ) ad, i this case, the ifiite product is said to coverge, otherwise the product is said to diverge. It follows from the relatio p = +a )p that ) p = p = + a )p = + a p, that is, if the ifiite product coverges, the a = 0, which, therefore, comprises a ecessary coditio for covergece. A aalysis of covergece of a ifiite product ca be reduced to aalyzig covergece of a series: [ ] p = exp k= l + a k ) exps so that from cotiuity of the expoetial fuctio it follows that [ ] p = p = exp S provided the it of the sequece of partial sums S exists. Thus, the existece of the it of {S } is a sufficiet coditio for the value p of

2. SERIES 9 the ifiite product to exist. The ifiite product + a ) is said to coverge absolutely if the series l + a ) coverges absolutely. Theorem 2.4. Absolute covergece of ifiite products) I order for a ifiite product + a ) to coverge absolutely, it is ecessary ad sufficiet that the series a coverges absolutely. Oe has to prove that the series l + a ) coverges absolutely if ad oly if the series a coverges absolutely. This objective is achieved by meas of the compariso test. I other words, the covergece of a implies the covergece of l + a ) ad the divergece of a implies the divergece of l + a ). First, ote that a 0 as 0 meas that there are oly fiitely may terms a that are greater tha ay ε > 0. I particular, a < 2, > N for some iteger N. Usig the Taylor expasio of the log fuctio for > N: l + a ) a = a 2 2 + a2 3 a3 4 + 2 a + 3 a 2 + 4 a 3 + 2 2 This iequality ca rewritte i the form a + a 2 + a 3 + ) 2 + 2 + ) 2 2 + = 3 2 2 a l + a ) 3 2 a By the compariso test it follows from the first iequality that the divergece of a implies the divergece of l + a ), ad, from the secod iequality, that the covergece of a implies the covergece of la + a ), as required. 2.4. Rearragemets of terms i ifiite products. If {a } is a rearragemet of {a }, the the product a is called a rearragemet of the product a. If the series a coverges absolutely, the the series l + a ) coverges absolutely. Therefore the sequeces of partial sums of the series l + a ) ad its rearragemet l + a ) coverge to the same umber. I tur, this implies that the value of

20. THE THEORY OF CONVERGENCE the product + a ) does ot deped o the order of terms i the product, provided a coverges absolutely. Cosider three ifiite products: ) ) ) z2 z2 z2 π 2 2 2 π 2 3 2 π ) ) ) 2 ) z [ π z π + z π ) e z π ] [ + z π z 2π ) e z π + z ] [ 2π z 2π At a glace they seem idetical because ) e z 2π ] [ + z ) ] e z 2π 2π z2 2 π = z ) + z ) [ = z ) ] [ e z 2 π + z ) ] e z π π π π π However, the first product coverges absolutely because the series z 2 2 π 2 = z2 π 2 coverges absolutely, while the secod product coverges oly coditioally because the series 2 z π + z π z 2π + z 2π + does ot coverge absolutely because = ). Therefore the first two products are equal if the order of terms i the secod product is as give, ad a rearragemet of it ca have a differet value or eve diverge. The third product also coverges absolutely because its 2m )th ad mth terms of the correspodig sequece {a } have the form z ) e ± z mπ = πm z πm ) = O m 2 ) ± z )) πm + O m 2 ad the series /m 2 coverges. So, for ay rearragemets the first ad third products have the same value as they coverge absolutely. A similar aalysis show that the product [ z ) ] e z c +

3. SEQUENCES AND SERIES OF FUNCTIONS 2 coverges absolutely for all complex z for ay c that is ot equal to a egative iteger, while the product [ ) ] z =2 coverges absolutely for all z >. 2.5. Evaluatio of ifiite products. I geeral, a evaluatio of ifiite products is a difficult task. The followig theorem gives a expasio of aalytic fuctios ito ifiite products. Theorem 2.5. Weierstrass) Let a aalytic fuctio fz) of a complex variable z have simple zeros that form a sequece {a } of o-zero elemets such that a as. The ) f 0) ) ] fz) = f0)exp [ za e z a f0) where the product coverges absolutely i ay disk z < K. For example, the fuctio siz)/z is aalytic i the etire complex plae the correspodig power series has ifiite radius of covergece) ad has simple zeros at z = π, where is a o-zero iteger, which ca be arraged ito a sequece {a } = {π, π, 2π, 2π,...} so that a as. It follows from the Weierstrass theorem that the third ifiite product cosidered i the previous sectio coverges absolutely to siz)/z ad, hece, so does the first oe. The secod product coverges to siz)/z i the give order of terms. 3. Sequeces ad series of fuctios I what follows, it is assumed that x R N, ad x a meas that the Euclidea distace betwee x ad a teds to zero, x a 0. 3.. Poitwise covergece. Cosider a sequece of fuctios u x) real or complex valued), =, 2,... The sequece {u } is said to coverge poitwise to a fuctio u o a set D if u x) = ux), x D. Similarly, the series u x) is said to coverge poitwise to a fuctio ux) o D R N if the sequece of partial sums coverges poitwise to

22. THE THEORY OF CONVERGENCE ux): S x) = u k x), k= S x) = ux), x D. The most importat questio: does the it fuctio or the sum) ux) iherit some properties of the terms of the sequece or the series)? I particular, If all terms u are cotiuous fuctios, is the it fuctio or the sum ux) cotiuous so that u x) = u x), x a x a u x) = u x)? x a x a If all terms u are differetiable fuctios, is the it fuctio or the sum ux) differetiable, ad, if affirmative, are the equatios d dx u d x) = dx u x), d d u x) = dx dx u x), valid? If all terms u are Riema) itegrable o a iterval [a, b], is the it fuctio or the sum ux) Riema itegrable so that b b ) u x)dx = u x) dx, a a b b ) u x)dx = u x) dx? a The aswer to all these questios is egative i geeral. A sequece or series of smooth fuctios with ifiitely may derivatives) may coverge to a fuctio that is ot cotiuous, or ot differetiable, or ot itegrable so that the stated equatios are geerally false. 3.2. Examples of coverget fuctioal sequeces ad series. Put u x) = a x 2, x R, = 0,, 2,... + x 2 )

3. SEQUENCES AND SERIES OF FUNCTIONS 23 ad cosider the series u x). The sequece of partial sums is easy to fid usig the geometric sum: ) k S x) = x 2 = x 2 + + x 2 + x 2 ). Therefore k=0 { 0, x = 0 S x) = ux) = + x 2, x 0 The terms u x) are ratioal fuctios defied o R ad, hece, differetiable ay umber of times, whereas the sum is ot eve cotiuous at x = 0. Cosider the fuctio ux), x R, that is defied as the it of the double sequece: u m x) = u m x) = [cosπxm!)] 2, ux) = u mx). m If x = p/q is a ratioal umber, the cos 2 πxm!) = for m q so that u m x) = ad ux) =. If x is irratioal, the the umber xm! caot be a iteger for ay m so that cos 2 πxm!) < for ay m so that u m x) = 0 ad, hece, ux). Thus, the it fuctio is the Dirichlet fuctio: {, x Q ux) = 0, x / Q where Q deotes the set of all ratioal umbers. The terms u m x) are smooth fuctios, whereas the it fuctio ux) is cotiuous owhere ad ot Riema itegrable o ay iterval. Cosider a uiform partitio of a iterval [a, b]: x k = a + k x, k = 0,, 2,..., N, x = b a)/n. Put M k = sup Ik ux) ad m k = if Ik ux), where I k = [x k, x k ]. Recall that ux) is Riema itegrable o [a, b] if the its of the upper ad lower sums, N N U N = M k x, L N = m k x exist ad are equal: k= U N = N b a k= ux)dx = N.L N Ay partitio iterval I k cotais ratioal ad irratioal umbers so that M k = ad m k = 0 for the Dirichlet fuctio. Therefore U N = b a ad L N = 0, that is, the its of the upper ad lower sums

24. THE THEORY OF CONVERGENCE exist but are ot equal, which meas that the Dirichlet fuctio is ot Riema itegrable. Eve if the it fuctio ux) of a sequece {u } happes to be differetiable, the sequece {u } of the derivatives does ot ecessarily coverge to the derivative u the operatios of or ) ad d/dx are ot commutative). Cosider a sequece u x) = six)/. It coverges poitwise to ux) = 0 for all x R. A costat fuctio is differetiable everywhere, u x) = 0. However, the it of the sequece of derivatives u x) = cosx) does ot exist ad, hece, u x) u x). The ext importat questio is: What are coditios uder which the it fuctio or the sum of a series iherits some properties of the terms of a sequece or series? I particular, uder what coditios ca the order of operatios or ) ad d/dx ad/or b ) be chaged? a 3.3. Uiform covergece. Defiitio 3.3. Uiform covergece) A sequece of fuctios {u } is said to coverge uiformly to a fuctio u o a set D R N if sup u x) ux) = 0 D Similarly, a series u x) coverges uiformly to ux) o D if the sequece of partial sums coverges uiformly to ux) o D. For example, the sequece u x) = six)/ coverges uiformly to ux) = 0 o R because u x) ux) = six) 0 as. The series x 2 / + x 2 ) does ot coverge uiformly o ay iterval whose closure cotais x = 0. Ideed, as show i the previous sectio S x) ux) = sup S + x 2 ) + x) ux) = I S x) ux) = 0 sup I for ay iterval closed or ope) I for which x = 0 is a edpoit. However, the series coverges uiformly for 0 < a x < or <

3. SEQUENCES AND SERIES OF FUNCTIONS 25 x a < 0 because sup I S x) ux) = = 0, a 0. + a 2 ) + Defiitio 3.4. A series u x) = ux) is said to coverge uiformly to ux) o a set D if the sequece of partial sums of the series coverges uiformly to ux) o D. Theorem 3.6. Sufficiet coditios for uiform covergece) Suppose that u x) M for all x D ad =, 2,... If the series of the upper bouds M < coverges, the the series u x) coverges uiformly o D. For example, a power series =0 a z coverges uiformly o ay disk z b < R, where R is the radius of covergece of the power series. Ideed, put u z) = a z. The u z) = a z a b = M, z b. The series of upper bouds coverges M < by the root test: sup M = b sup a = b R < where the defiitio of the radius of covergece has bee used. Therefore the power series coverges uiformly i ay disk of radius less tha the radius of covergece. Theorem 3.7. Cauchy criterio for uiform covergece) A sequece of fuctios {u } coverges to a fuctio u uiformly o a set D if ad oly if for ay ε > 0 oe ca fid a iteger N such that, m > N implies that u x) u m x) ε x D The differece with the ordiary Cauchy criterio for covergece is that the iteger N here depeds oly o ε, that is, for all x i D oe ca fid the same N, give ε > 0. 3.4. Uiform covergece ad cotiuity. Theorem 3.8. Suppose that a sequece {u x)} coverges uiformly to ux) o a set D R N. Let y be a it poit of D so that u x) = A, =, 2,.... x y

26. THE THEORY OF CONVERGENCE The the sequece of its {A } coverges ad A = u x) x y If y D so that u y) is defied, the the stated theorem implies that the it of a uiformly coverget sequece of cotiuous fuctios is a cotiuous fuctio: u x) = u y). x y For example, the fuctio defied by the Fourier series e ix ux) = + 2 is cotiuous everywhere. Ideed, =0 u x) = eix 2 u x) = + 2 2, ad the series of upper bouds coverges 2 <. Therefore the series u x) coverges uiformly for all x. Sice u x) are cotiuous everywhere, the sum of the Fourier series is also a cotiuous fuctio. 3.5. Uiform covergece ad differetiatio. Theorem 3.9. Let {u } be a sequece of differetiable fuctios o [a, b]. Suppose that the sequece {u x 0 )} coverges for some x 0 [a, b] ad the sequece of derivatives {u x)} coverges uiformly o [a, b]. The the sequece {u } coverges uiformly to some ux) o [a, b], the it fuctio ux) is differetiable o [a, b] ad u x) = u x) ) = u x). For example, the sum of the series si 2 x) ux) = 2 + 3 is a differetiable fuctio everywhere. Ideed, each term of the series is differetiable everywhere. If x = 0, the the sum is u0) = 0. The series of the derivatives coverges uiformly everywhere because the series of upper bouds u x) = si2x) 2 + 3 2 + 3 2

3. SEQUENCES AND SERIES OF FUNCTIONS 27 coverges <. Therefore 2 u x) = si2x) 2 + 3. The sum of a power series is ifiitely may times differetiable fuctio i the disk z < R where R is the radius of covergece. Ideed, the series of derivatives coverges uiformly i ay disk z b < B because the correspodig series of upper bouds u z) = a z a b = M coverges, M <, by the root test: sup Therefore M = b sup a /b = b R <. u z) = u z) = =0 a z Similarly, the series of the kth derivatives ) k)a z k =k coverges uiformly i z b < R by the same reaso. Therefore it coverges to the kth derivative of the sum u k) z). The coclusio holds for ay k. So, uz) is differetiable ifiitely may times ad ay derivative is obtaied by term-by-term differetiatio of the power series. A fuctio defied as it of a sequece or the sum of series may be cotiuous everywhere but owhere differetiable! Let fx) = x for x. Let us exted f to R by periodicity: fx + 2) = fx) Evidetly, fx) is cotiuous everywhere ad bouded: fx), x R. Defie the fuctio ux) o R by the series ) 3 ux) = f4 x). 4 =0

28. THE THEORY OF CONVERGENCE This series of cotiuous terms coverges uiformly o R because the series of upper bouds coverges: ) 3 f4 x) 3 ) 3 4 4) = 4 < 4 as a geometric series. Therefore the sum ux) is cotiuous o R. However ux) is differetiable owhere. By defiitio u x) = δ 0 ux + δ) ux) δ This it does ot exist for ay x. If this it existed, the for ay sequece δ m covergig to 0 as m, the it =0 ux + δ m ) ux) m δ m must exist ad coicide with u x). Put δ m = ± 2 4 m where the sig is chose so that o iteger betwee lies betwee 4 m x ad 4 m x + δ m ) = 4 m x ±. Clearly, δ 2 m 0 as m. By the choice of δ m : f4 m x + δ m )) f4 m x) = f4 m x ± ) 2 f4m x) = ± 2 = 2 because the fuctio f has a costat slope, or, betwee ay two itegers, ad o iteger lies betwee 4 m x ad 4 m x + δ m ). Put γ = f4 x + δ m )) f4 x) δ m The it follows from the above property of f that γ m = 2 δ m = 4m. If > m, the umber 4 δ m is eve, ad by periodicity of f The ux + δ m ) ux) δ m = = = δ m =0 γ = 0, > m ) 3 f4 x + δ m )) 4 δ m =0 ) 3 f4 x + δ m )) f4 x) 4 ) 3 γ = 4 =0 =0 m =0 δ m 3 4 ) m γ = 3 m ) 3 f4 x) 4 ) 3 γ 4

3. SEQUENCES AND SERIES OF FUNCTIONS 29 This sequece diverges as m. Ideed, usig the iequality A B A B m 3m =0 ) 3 γ 4 3m m =0 =0 3 4 ) γ m 3 m 3 m = 2 3m ) as m. So, the lower boud diverges as m ad this implies that the it that defies the derivative u x) does ot exist. Thus, u x) does ot exist for ay x R. 3.6. Uiform covergece ad Riema itegratio. Theorem 3.20. Let {u } be a sequece of fuctios that are Riema itegrable o [a, b]. If the sequece coverges to ux) uiformly o [a, b], the the it fuctio ux) is Riema itegrable o [a, b] ad b b ux)dx = u x)dx. a a I particular, a fuctio defied by a coverget power series ca be itegrated term-by-term ad the resultig series coverges to a itegral of the sum i ay disk whose radius is less tha the radius of covergece of the origial series.