ENGI Series Page 601


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1 ENGI Series Page Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series, pseries 6.04 Tests for Covergece: compariso ad limit compariso tests 6.05 Tests for Covergece: alteratig series; absolute ad coditioal covergece 6.06 Tests for Covergece: ratio test 6.07 Power Series, radius ad iterval of covergece 6.08 Taylor ad Maclauri Series, remaider term 6.09 Biomial Series 6.10 Itroductio to Fourier Series Appedix: 6.A Itegral Test [ot examiable; for referece oly]
2 ENGI Sequeces Page Sequeces; geeral term, limits, covergece A sequece is a set of related objects that all follow the same rule. progressio from curret ad previous elemets to the ext elemet. There is a logical Example The alphabet: { a, b, c,..., y, z } The rule is: The th elemet a = (the th letter of the alphabet) Example The set of all atural umbers = { 1, 2, 3,... } Two ways to express the rule are: We are usually iterested i two features: A explicit form for the geeral term (ofte from a recurrece relatioship); The behaviour of the terms of a sequece as. Example The terms of a sequece form a arithmetic progressio if cosecutive terms differ by a commo ozero costat differece d : a a d a a d, with a iitial term a a Writig dow the first few terms of the sequece, The geeral term is therefore a
3 ENGI Sequeces Page 603 Example The terms of a sequece form a geometric progressio if cosecutive terms chage by a commo costat ratio r : a a 1 1 r a a r r 0, r 1, with a iitial term a1 a 0 Writig dow the first few terms of the sequece, The geeral term is therefore a The sequece alterates i sig if r < 0 The sequece coverges to a limit of zero if If r = 1 the lim a
4 ENGI Sequeces Page 604 The power sequece r 1 r, 2 r, 3 r, 4 r, coverges to 0 for r < 0, coverges to 1 for r = 0 ad diverges for r > 0.
5 ENGI Sequeces Page 605 Example Fid the geeral term of this sequece ad determie whether it coverges ,,,,, Algebra of Sequeces: Give two coverget sequeces pa a with limit value A ad b with limit value B, qb is a coverget sequece with limit value (pa + qb) for all costats p, q; ab is a coverget sequece with limit value AB; a/ b is a coverget sequece with limit value A/B (provided that B 0 ); If f (x) is a cotiuous fuctio with lim f x L a f the a coverges with limit value L ; If a c b ad if c coverges with limit value C the A C B. ad if
6 ENGI Sequeces Page 606 Example Fid the limit of the sequece si. The Race to Ifiity There is a hierarchy of fuctioal forms that all diverge to ifiity with icreasig : c! a a 1 c 0 logb b 1 for all sufficietly large. Example ! lim 0 because goes to ifiity faster tha! OR the sequece is positive ad! Therefore all terms are bouded above by the terms of a sequece that coverges to 0.
7 ENGI Sequeces Page 607 If every term of a sequece is o less tha all of its predecessors a1 a2 a3 a4, the the sequece is a icreasig sequece. If every term of a sequece is o greater tha all of its predecessors a1 a2 a3, the the sequece is a decreasig sequece. A sequece is mootoic if ad oly if it is either icreasig or decreasig. If a U for some costat value U, the the sequece for some costat value L, the the sequece If a L A sequece is bouded if it is both upper ad lower bouded. a is upperbouded. a is lowerbouded. Every upperbouded icreasig sequece is coverget. Every lowerbouded decreasig sequece is coverget. Every bouded mootoic sequece is coverget.
8 ENGI Series Page Series The partial sum S of a sequece is the sum of the first terms of that sequece: S a a a a a k k 1 This partial sum is also a fiite series. The sum of a ifiite series is the sum of all terms i the associated ifiite sequece, S a 1 a 2 a a 3 k lim S. k 1 Example ad fid the limit of the sequece of partial sums. Fid the first three partial sums of the sequece, 1, 2, 3,
9 ENGI Series Page 609 Divergece Test If the geeral term of a series does ot coverge to zero, the the series diverges. lim a 0 a diverges (the sum does ot exist) 1 Example Prove that all arithmetic series 1 a d diverge, (where d 0 ). 1 However, the coverse of the divergece test is false. Some diverget series do have geeral terms that coverge to zero, or equivaletly: lim a 0 does ot guaratee that (except for alteratig series, sectio 6.05 below). a coverges 1
10 ENGI Series Page 610 Example Show that the harmoic series 1 diverges. 1 The divergece of the harmoic series is very slow. 4 terms are eeded for the partial sum to exceed terms are eeded for the partial sum to exceed terms are eeded for the partial sum to exceed 8. The sum of the first billio (10 9 ) terms is still well uder 25. Yet the sum of the complete series does ot exist (diverges to ifiity)!
11 ENGI Stadard Series Page Stadard Series: telescopig series, geometric series, pseries Example Fid the exact sum of the series S This is a example of a telescopig series. The simplest such series have geeral terms similar to a f f 1 Such series ofte ivolve the use of partial fractios..
12 ENGI Stadard Series Page 612 Example Fid the exact sum of the series S
13 ENGI Stadard Series Page 613 Example (cotiued) S = Collectig the survivig terms from this telescopig series, we fid that
14 ENGI Stadard Series Page 614 Geometric Series Each term i a geometric series is a costat multiple r (the commo ratio) of the immediately precedig term, except for the [ozero] first term a. Quotig the geeral term from the geometric progressio o page 6.03, the geometric series is Applyig the divergece test: 1 or 1 0 S ar ar
15 ENGI Stadard Series Page 615 Example (Example revisited) Evaluate S Example Fid the sum of the power series f x x x x x 2 3 4
16 ENGI Stadard Series Page 616 pseries Aother importat family of stadard series is the pseries (also kow as hyperharmoic series): 1 1 The divergece test establishes divergece for p < 0. It ca be show [sectio 6.A] that pseries coverge for all p > 1 ad diverge for all p < 1. The case p = 1 is the harmoic series, for which Example established divergece. p I 1741 Leohard Euler proved that The removal or isertio of a fiite umber of fiite terms does ot affect the covergece of a series. If S a coverges, the so does a a1 a2 a a1 a2 S 3 provided that a 1 ad a 2 are fiite., 1 3
17 ENGI Compariso Tests Page Tests for Covergece: compariso ad limit compariso tests If all terms a i a series are positive, the the sequece of partial sums S is ecessarily icreasig. If aother series b is coverget with a sum B ad if a b, the 1 B is a upper boud to S Recall that a icreasig sequece that has a upper boud is coverget [page 6.07] S must coverge to a value < B a coverges to some value A < B. 1 This is the basis of the compariso test. Example Is the series coverget? [This is Example , for which the exact value of the sum was foud by telescopig the series.]
18 ENGI Compariso Tests Page 618 Example Is the series 1 l coverget? 2 Put very briefly, the essece of the compariso test is: A coverget ceilig forces covergece: 0 a b ad b coverges a coverges. A diverget floor forces divergece: a b 0 ad b diverges a diverges.
19 ENGI Compariso Tests Page 619 Limit Compariso Test All terms a ad b must be oegative ad L is some fiite costat. a lim L 0 ad b coverges a coverges. b a lim 0 ad b diverges a diverges. b The referece series b 1 is ofte the geometric series or a pseries. Example l Is the series coverget? 3 8 3
20 ENGI Compariso Tests Page 620 Example Is the series coverget? 3 9 1
21 ENGI Alteratig Series Test Page Tests for Covergece: alteratig series; absolute ad coditioal covergece Alteratig Series Test If cosecutive terms of a series alterate i sig, the a simpler test for covergece is available. a If 1 0 a ad lim a 0 the a coverges. 1 Example Is the alteratig harmoic series a coverget? 1 1 Note that if we take the absolute values of the terms i the alteratig harmoic series, the we obtai the harmoic series 1, which diverges
22 ENGI Alteratig Series Test Page 622 Absolute Covergece If ad oly if the series the the series 1 a If ad oly if the series the the series 1 a a coverges, 1 is absolutely coverget. a coverges but the series a diverges, 1 1 is coditioally coverget. The alteratig harmoic series is a example of a coditioally coverget series. If a series is absolutely coverget, the the alteratig series test is a waste of time. Aother test to establish the covergece of 1 a is absolutely coverget i oe step. 1 a automatically establishes that If a series is coditioally coverget, the the alteratig series test is ofte required to establish covergece of a, but aother test is required o a to complete the 1 1 proof of coditioal covergece. Therefore test the covergece of a first! 1 Absolute covergece allows the algebra of fiite series to be exteded to ifiite series. I particular, the order of a ifiite umber of terms may be chaged without affectig the sum of the series. The derivative of a absolutely coverget series is the sum of the derivatives of the idividual terms. This is o loger ecessarily true for coditioally coverget series. For example, l 2 but 1 l
23 ENGI Alteratig Series Test Page 623 Example Ivestigate the covergece of the alteratig pseries a 1 p p 3 p 4 p This is obviously a alteratig series. Example Is 1 absolutely coverget, coditioally coverget or diverget?
24 ENGI Alteratig Series Test Page 624 Approximatio Errors For ay coverget alteratig series, the error i approximatig the sum S of the etire series caused by evaluatig the partial sum S of just the first terms is S S a 1 Example Estimate S 0 1 correct to four decimal places. 2! The sigs of the terms of this series are clearly alteratig ad this series coverges (by the alteratig series test). 1 lim 0 2! The error caused by usig S to estimate S is
25 ENGI Ratio Test Page Tests for Covergece: ratio test Amog the most useful tests for series covergece is the ratio test: a If lim 1 1 the a a If lim 1 1 the a 1 1 a a is absolutely coverget. is diverget. a If lim 1 1 the the test fails ad there is o iformatio o series covergece. a The ratio test is most useful whe the geeral term icludes a expoetial or factorial factor. The test fails whe the geeral term is a algebraic fuctio (terms of the form costat i the umerator ad/or deomiator) oly. Example Is the series diverget? a 1 1! absolutely coverget, coditioally coverget or
26 ENGI Ratio Test Page 626 Note: Proof that 1 lim 1 e :
27 ENGI Ratio Test Page 627 Example Is the series diverget? a 0 0 x! absolutely coverget, coditioally coverget or There is a test for absolute covergece kow as the root test, which is a close relative of the ratio test. However, we shall ot use the root test i ENGI 3425.
28 ENGI Power Series Page Power series, radius ad iterval of covergece A power series with cetre x = c is of the form bd d b 2b 3b f x a x c x c a a x c a x c a x c 0 where b ad d are real costats ad b A commo choice of parameters for power series is b = 1 ad d = 0: f x a x c a a x c a x c a x c u ow represet the geeral term b d Let a x c ad let us apply the ratio test for covergece of a power series: b b d bd a x c a u a x c a x c u u lim 1 b x c lim 1 u a The series is absolutely coverget wheever a b where R is the radius of covergece The iterval of covergece I icludes c R x c R ad may iclude oe or both edpoits, which must be tested separately. I is also the domai of the fuctio f (x). Also ote that f (c) = 0 a. All power series are absolutely coverget at the cetre, eve if R = 0. If R, the the power series is absolutely coverget for all x.
29 ENGI Power Series Page 629 Example Determie the radius of covergece ad iterval of covergece for the power series f x 0 x 2 3
30 ENGI Power Series Page 630 Example Determie the radius of covergece ad iterval of covergece for the power series x 1 f x 3 1
31 ENGI Taylor ad Maclauri series Page Taylor ad Maclauri Series A fuctio f (x) that is represeted by a power series (usig oly oegative iteger powers of x c ) is a Taylor series: f x a x c f c a 0 0 a f c Assumig that the series exists ad is absolutely coverget, the f x a x c 0 1 f c 0 1a 0 0 a 1 1 f c 1 1 f x a x c 0 2 f c a 0 a 2 2 f c f x a x c 0 3 f c f c a3 a3 ad so o, so that the geeral term is a f c ad the Taylor series for f (x) is! c f f x x c! The ratio test ca determie the radius of covergece R.
32 ENGI Taylor ad Maclauri series Page 632 Example Fid the Taylor series for f x l x about x = 1.
33 ENGI Taylor ad Maclauri series Page 633 Example Prove L Hôpital s rule for the case of a (0/0) idetermiacy. Let fuctios f (x) ad g (x) be such that f (a) = g (a) = 0. f a The is a 0 type of idetermiacy. g a 0 Represet both fuctios by their Taylor series about x = a, the
34 ENGI Taylor ad Maclauri series Page 634 Maclauri Series A Taylor series with a cetre at x = c = 0 is a Maclauri series. Example x Fid the Maclauri series for f x e ad fid its iterval of covergece.
35 ENGI Taylor ad Maclauri series Page 635 Example Fid the Maclauri series for f x cos x ad fid its iterval of covergece.
36 ENGI Taylor ad Maclauri series Page 636 Example (Alterative Solutio) Use the Euler expressio for the cosie fuctio i terms of the expoetial fuctio: e j cos jsi ad cos
37 ENGI Taylor ad Maclauri series Page 637 Example Use the Maclauri series for the cosie fuctio to fid the Maclauri series for the sie fuctio. The Maclauri series for f (x) = si x may also be foud directly, through repeated differetiatios, or via the Euler equatio ad the Maclauri series for the expoetial fuctio (as i Example for the cosie fuctio) or by differetiatio of cos x.
38 ENGI Taylor ad Maclauri series Page 638 Remaider Term The practical use of Taylor (ad Maclauri) series arises whe the partial sum of the first few terms of the series T x is used to approximate the value of a oliear fuctio. The error caused by trucatig the series at the th term is the th remaider term R x : 1 1! f R x f x T x x c where is some umber betwee x ad c. 1 The Taylor series is welldefied oly if lim R x 0 c R x c R. Example x The Maclauri series for the expoetial fuctio f x e is e x 2 3 x x x 1 x k! 2! 3! k 0 k 1 1 e x f R x x 0 0 x 1! 1! Because factorial fuctios diverge to ifiity faster tha expoetial fuctios, 1 x lim R x e lim 0 1! ad the Maclauri series is therefore welldefied. x The error caused by approximatig f x e by the quadratic 3 a 1 2 x T2 x 1 x o 2! ae the iterval (0 < x < a) is at most. 6 For a = 0.1, this maximum error is less tha , with a maximum relative error of uder 0.03%. However, for a = 10, the maximum error is substatial: more tha May more terms eed to be take to maitai accuracy, the further away from the cetre oe goes. Note that is the equatio of the taget lie to y f x y T x 1 at x = c.
39 ENGI Biomial Series Page Biomial Series A special case of Maclauri series arises for f x 1 x.
40 ENGI Biomial Series Page 640 Summary: x 1 1 k k k! x k Iterval of covergece: I for = 0 or [ the series termiates at k = ] I = [ 1, 1 ] for > 0 ad ot a iteger, with absolute covergece at x = ±1. I = ( 1, 1 ] for 1 < < 0 with coditioal covergece at x = +1. I = ( 1, 1 ) for < 1 with divergece at x = ±1.
41 ENGI Biomial Series Page 641 Example Fid the Maclauri series for f x 1 2x ad its iterval of covergece. I the biomial expasio, for "x" read "2x" ad for "" read "½".
42 ENGI Fourier Series Page Fourier Series This sectio offers a very brief overview of [discrete] Fourier series. Some majors will explore Fourier series ad trasforms i much greater detail i subsequet semesters. The Fourier series of f x o the iterval ( L, L) is where ad a0 x x f x acos bsi 2 1 L L L 1 x a f x dx L L L L cos, 0,1, 2, 3, 1 x b f x dx L L L si, 1, 2, 3, The a, b are the Fourier coefficiets of f x. Note that the cosie fuctios (ad the fuctio 1) are eve, while the sie fuctios are odd. If f x is eve ( f x f x for all x), the b 0 for all, leavig a Fourier cosie series (ad perhaps a costat term) oly for f x. If f x is odd ( f x f x for all x), the a 0 for all, leavig a Fourier sie series oly for f x. Note that a a g x dx 0 a 0 if g x is odd 2 g x dx if g x is eve with the multiplicatio table odd fuctio eve fuctio odd fuctio eve odd eve fuctio odd eve
43 ENGI Fourier Series Page 643 Example Expad f x 0 x 0 x 0 x i a Fourier series.
44 ENGI Fourier Series Page 644 Example (Additioal Notes also see " The first few partial sums i the Fourier series 11 1 f x cos x si x 2 x 4 1 are S0 4 S 2 1 cos si 4 x x S cos si si 2 4 x x 2 x S cos si si 2 cos3 si 3 4 x x 2 x 9 x 3 x ad so o. The graphs of successive partial sums approach f (x) more closely, except i the viciity of ay discotiuities, (where a systematic overshoot occurs, the Gibbs pheomeo).
45 ENGI Fourier Series Page 645 Example Fid the Fourier series expasio for the stadard square wave, 1 1 x 0 f x 1 0 x 1 The graphs of the third ad ith partial sums (cotaiig two ad five ozero terms respectively) are displayed here, together with the exact form for f (x), with a periodic extesio beyod the iterval ( 1, +1) that is appropriate for the square wave. y = S 3 (x)
46 ENGI Fourier Series Page 646 Example (cotiued) y S9 x Covergece At all poits limits lim x x o x x i ( L, L) where f x is cotiuous ad is either differetiable or the f o x ad lim f x both exist, the Fourier series coverges to x x o f x. At fiite discotiuities, (where the limits lim x x o f x ad lim f x x x o both exist), the f xo f xo Fourier series coverges to, 2 f x lim f x ad f x lim f x ). (usig the abbreviatios o o xx o xx o f (x) ot cotiuous cotiuous but cotiuous ad at x x ot differetiable differetiable I all cases, the Fourier series at x x coverges to o o f x f x o o (the red dot). 2
47 ENGI Fourier Series Page 647 HalfRage Fourier Series A Fourier series for f (x), valid o [0, L], may be costructed by extesio of the domai to [ L, L]. A odd extesio leads to a Fourier sie series: where L 0 x f x b si L 1 2 x b f x dx L L si, 1, 2, 3, A eve extesio leads to a Fourier cosie series: where a0 x f x a cos 2 L L x a f x dx L L cos, 0,1, 2, 3, ad there is automatic cotiuity of the Fourier cosie series at x = 0 ad at x = L.
48 ENGI Fourier Series Page 648 Example Fid the Fourier sie series ad the Fourier cosie series for f (x) = x o [0, 1]. f (x) = x happes to be a odd fuctio of x for ay domai cetred o x = 0. The odd extesio of f (x) to the iterval [ 1, 1] is f (x) itself. Evaluatig the Fourier sie coefficiets, This fuctio happes to be cotiuous ad differetiable at x = 0, but is clearly discotiuous at the edpoits of the iterval (x = 1). Fifth order partial sum of the Fourier sie series for f (x) = x o [0, 1]
49 ENGI Fourier Series Page 649 Example (cotiued) The eve extesio of f (x) to the iterval [ 1, 1] is f (x) = x. Evaluatig the Fourier cosie coefficiets,
50 ENGI Fourier Series Page 650 Example (cotiued) Third order partial sum of the Fourier cosie series for f (x) = x o [0, 1] Note how rapid the covergece is for the cosie series compared to the sie series. for cosie series ad y S x y S x 3 for sie series for f (x) = x o [0, 1] 5
51 ENGI A Itegral Test Page A Itegral Test [for referece oly ot examiable i this course] If f (x) is a cotiuous fuctio with positive values that are decreasig for all x > k,, 1, 2, ad the improper itegral lim a f k k k exists as a fiite real umber, the the series k k k k a coverges ad k f x dx a a f x dx If the improper itegral does ot have a fiite value, the the series diverges. L L f x dx x k Proof: Examie the area betwee x = i 1 ad x = i uder the curve y = f (x) : Both rectagles have width oe uit. The smaller rectagle has height f (i) ad area smaller tha that uder the curve. The larger rectagle has height f (i 1) ad area larger tha that uder the curve. Therefore i 1 f i f x dx 1 f i 1 i1 This iequality is true for all areas from i 1 = k owards. Addig all of these areas together, we fid The left side ca be rearraged as k 1 f i f x dx f i k i k1 i k1 f x dx f i f k f i f k f x dx i k i k The right side ca be rearraged as 1 f i f i f x dx f x dx f i k k i k1 i k i k k
52 ENGI A Itegral Test Page 652 Also ote that the i th term of the series is a i f i k k k. It the follows that k f x dx a a f x dx The covergece or divergece of the series is therefore liked completely to the covergece or divergece of the improper itegral. The double iequality also allows upper ad lower bouds to be evaluated for the sum of a coverget series whose terms form a sequece draw from a positive, cotiuous ad decreasig fuctio f (x). The covergece of the pseries ca be established by a combiatio of the divergece ad itegral tests: p 0 1 lim p 1 p 0 0 p 0 the pseries diverges for p < 0. 1 Whe p > 0 the fuctio f x p x is positive, cotiuous ad decreasig o [1, ) so that we may use the itegral test. p1 x 1 lim p 1 p p 1 p 1 x dx 1 p p 1 lim l x p Therefore the pseries p coverges for p > 1 ad diverges otherwise. 1 Aother example of the itegral test arises i showig that 1 p coverges if ad oly if p > 1 [ details omitted]. 2 l END OF CHAPTER 6
ENGI Series Page 501
ENGI 344 5 Series Page 501 5. Series Cotets: 5.01 Sequeces; geeral term, limits, covergece 5.0 Series; summatio otatio, covergece, divergece test 5.03 Series; telescopig series, geometric series, pseries
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