ENGI Series Page 6-01
|
|
- Abigayle Matthews
- 5 years ago
- Views:
Transcription
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, p-series 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 o-zero 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 6-03 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 6-04 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 6-05 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 6-06 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 6-07 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 upper-bouded. a is lower-bouded. Every upper-bouded icreasig sequece is coverget. Every lower-bouded 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 6-09 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 6-10 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, p-series 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 6-12 Example Fid the exact sum of the series S
13 ENGI Stadard Series Page 6-13 Example (cotiued) S = Collectig the survivig terms from this telescopig series, we fid that
14 ENGI Stadard Series Page 6-14 Geometric Series Each term i a geometric series is a costat multiple r (the commo ratio) of the immediately precedig term, except for the [o-zero] 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 6-15 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 6-16 p-series Aother importat family of stadard series is the p-series (also kow as hyperharmoic series): 1 1 The divergece test establishes divergece for p < 0. It ca be show [sectio 6.A] that p-series 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 6-18 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 6-19 Limit Compariso Test All terms a ad b must be o-egative 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 p-series. Example l Is the series coverget? 3 8 3
20 ENGI Compariso Tests Page 6-20 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 6-22 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 6-23 Example Ivestigate the covergece of the alteratig p-series 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 6-24 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 6-26 Note: Proof that 1 lim 1 e :
27 ENGI Ratio Test Page 6-27 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 6-29 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 6-30 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 o-egative 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 6-32 Example Fid the Taylor series for f x l x about x = 1.
33 ENGI Taylor ad Maclauri series Page 6-33 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 6-34 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 6-35 Example Fid the Maclauri series for f x cos x ad fid its iterval of covergece.
36 ENGI Taylor ad Maclauri series Page 6-36 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 6-37 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 6-38 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 o-liear 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 well-defied 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 well-defied. 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 6-40 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 6-41 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 6-43 Example Expad f x 0 x 0 x 0 x i a Fourier series.
44 ENGI Fourier Series Page 6-44 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 6-45 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 o-zero 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 6-46 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 6-47 Half-Rage 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 6-48 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 6-49 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 6-50 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 re-arraged 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 re-arraged 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 6-52 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 p-series ca be established by a combiatio of the divergece ad itegral tests: p 0 1 lim p 1 p 0 0 p 0 the p-series 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 p-series 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 5-01
ENGI 344 5 Series Page 5-01 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, p-series
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More information4x 2. (n+1) x 3 n+1. = lim. 4x 2 n+1 n3 n. n 4x 2 = lim = 3
Exam Problems (x. Give the series (, fid the values of x for which this power series coverges. Also =0 state clearly what the radius of covergece is. We start by settig up the Ratio Test: x ( x x ( x x
More informationAP 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
More informationSUMMARY 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
More informationMAT1026 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
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More information7 Sequences of real numbers
40 7 Sequeces of real umbers 7. Defiitios ad examples Defiitio 7... A sequece of real umbers is a real fuctio whose domai is the set N of atural umbers. Let s : N R be a sequece. The the values of s are
More informationMath 113 Exam 4 Practice
Math Exam 4 Practice Exam 4 will cover.-.. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for
More information(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
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationTesting for Convergence
9.5 Testig for Covergece Remember: The Ratio Test: lim + If a is a series with positive terms ad the: The series coverges if L . The test is icoclusive if L =. a a = L This
More informationINFINITE SEQUENCES AND SERIES
INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES I geeral, it is difficult to fid the exact sum of a series. We were able to accomplish this for geometric series ad the series /[(+)]. This is
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationINFINITE 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
More information62. 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
More informationSOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.
SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad
More informationMa 530 Infinite Series I
Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li
More informationAlternating Series. 1 n 0 2 n n THEOREM 9.14 Alternating Series Test Let a n > 0. The alternating series. 1 n a n.
0_0905.qxd //0 :7 PM Page SECTION 9.5 Alteratig Series Sectio 9.5 Alteratig Series Use the Alteratig Series Test to determie whether a ifiite series coverges. Use the Alteratig Series Remaider to approximate
More informationThe 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
More informationIn 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
More informationCHAPTER 1 SEQUENCES AND INFINITE SERIES
CHAPTER SEQUENCES AND INFINITE SERIES SEQUENCES AND INFINITE SERIES (0 meetigs) Sequeces ad limit of a sequece Mootoic ad bouded sequece Ifiite series of costat terms Ifiite series of positive terms Alteratig
More informationPhysics 116A Solutions to Homework Set #1 Winter Boas, problem Use equation 1.8 to find a fraction describing
Physics 6A Solutios to Homework Set # Witer 0. Boas, problem. 8 Use equatio.8 to fid a fractio describig 0.694444444... Start with the formula S = a, ad otice that we ca remove ay umber of r fiite decimals
More informationChapter 6 Overview: Sequences and Numerical Series. For the purposes of AP, this topic is broken into four basic subtopics:
Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals (which is what most studets
More information10.6 ALTERNATING SERIES
0.6 Alteratig Series Cotemporary Calculus 0.6 ALTERNATING SERIES I the last two sectios we cosidered tests for the covergece of series whose terms were all positive. I this sectio we examie series whose
More informationCarleton College, Winter 2017 Math 121, Practice Final Prof. Jones. Note: the exam will have a section of true-false questions, like the one below.
Carleto College, Witer 207 Math 2, Practice Fial Prof. Joes Note: the exam will have a sectio of true-false questios, like the oe below.. True or False. Briefly explai your aswer. A icorrectly justified
More informationPractice 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,
More informationPLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 3. (a) (b) (c) (d) (e) 5. (a) (b) (c) (d) (e) 7. (a) (b) (c) (d) (e)
Math 0560, Exam 3 November 6, 07 The Hoor Code is i effect for this examiatio. All work is to be your ow. No calculators. The exam lasts for hour ad 5 mi. Be sure that your ame is o every page i case pages
More informationChapter 7: Numerical Series
Chapter 7: Numerical Series Chapter 7 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals
More informationMIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More informationSeries Review. a i converges if lim. i=1. a i. lim S n = lim i=1. 2 k(k + 2) converges. k=1. k=1
Defiitio: We say that the series S = Series Review i= a i is the sum of the first terms. i= a i coverges if lim S exists ad is fiite, where The above is the defiitio of covergece for series. order to see
More informationAns: a n = 3 + ( 1) n Determine whether the sequence converges or diverges. If it converges, find the limit.
. Fid a formula for the term a of the give sequece: {, 3, 9, 7, 8 },... As: a = 3 b. { 4, 9, 36, 45 },... As: a = ( ) ( + ) c. {5,, 5,, 5,, 5,,... } As: a = 3 + ( ) +. Determie whether the sequece coverges
More informationSequences. Notation. Convergence of a Sequence
Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it
More informationSeries 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
More informationMath 112 Fall 2018 Lab 8
Ma Fall 08 Lab 8 Sums of Coverget Series I Itroductio Ofte e fuctios used i e scieces are defied as ifiite series Determiig e covergece or divergece of a series becomes importat ad it is helpful if e sum
More informationMA131 - 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.............................
More informationf(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
More informationConvergence: nth-term Test, Comparing Non-negative Series, Ratio Test
Covergece: th-term Test, Comparig No-egative Series, Ratio Test Power Series ad Covergece We have writte statemets like: l + x = x x2 + x3 2 3 + x + But we have ot talked i depth about what values of x
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationINFINITE SERIES PROBLEMS-SOLUTIONS. 3 n and 1. converges by the Comparison Test. and. ( 8 ) 2 n. 4 n + 2. n n = 4 lim 1
MAC 3 ) Note that 6 3 + INFINITE SERIES PROBLEMS-SOLUTIONS 6 3 +
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationReview for Test 3 Math 1552, Integral Calculus Sections 8.8,
Review for Test 3 Math 55, Itegral Calculus Sectios 8.8, 0.-0.5. Termiology review: complete the followig statemets. (a) A geometric series has the geeral form k=0 rk.theseriescovergeswhe r is less tha
More informationChapter 6: Numerical Series
Chapter 6: Numerical Series 327 Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals
More informationx x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,
Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative
More informationExample 2. Find the upper bound for the remainder for the approximation from Example 1.
Lesso 8- Error Approimatios 0 Alteratig Series Remaider: For a coverget alteratig series whe approimatig the sum of a series by usig oly the first terms, the error will be less tha or equal to the absolute
More informationMTH 122 Calculus II Essex County College Division of Mathematics and Physics 1 Lecture Notes #20 Sakai Web Project Material
MTH 1 Calculus II Essex Couty College Divisio of Mathematics ad Physics 1 Lecture Notes #0 Sakai Web Project Material 1 Power Series 1 A power series is a series of the form a x = a 0 + a 1 x + a x + a
More informationFINALTERM EXAMINATION Fall 9 Calculus & Aalytical Geometry-I Questio No: ( Mars: ) - Please choose oe Let f ( x) is a fuctio such that as x approaches a real umber a, either from left or right-had-side,
More informationMath 341 Lecture #31 6.5: Power Series
Math 341 Lecture #31 6.5: Power Series We ow tur our attetio to a particular kid of series of fuctios, amely, power series, f(x = a x = a 0 + a 1 x + a 2 x 2 + where a R for all N. I terms of a series
More informationMath 132, Fall 2009 Exam 2: Solutions
Math 3, Fall 009 Exam : Solutios () a) ( poits) Determie for which positive real umbers p, is the followig improper itegral coverget, ad for which it is diverget. Evaluate the itegral for each value of
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationZ ß cos x + si x R du We start with the substitutio u = si(x), so du = cos(x). The itegral becomes but +u we should chage the limits to go with the ew
Problem ( poits) Evaluate the itegrals Z p x 9 x We ca draw a right triagle labeled this way x p x 9 From this we ca read off x = sec, so = sec ta, ad p x 9 = R ta. Puttig those pieces ito the itegralrwe
More informationMAS111 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
More informationTaylor Series (BC Only)
Studet Study Sessio Taylor Series (BC Oly) Taylor series provide a way to fid a polyomial look-alike to a o-polyomial fuctio. This is doe by a specific formula show below (which should be memorized): Taylor
More informationAn alternating series is a series where the signs alternate. Generally (but not always) there is a factor of the form ( 1) n + 1
Calculus II - Problem Solvig Drill 20: Alteratig Series, Ratio ad Root Tests Questio No. of 0 Istructios: () Read the problem ad aswer choices carefully (2) Work the problems o paper as eeded (3) Pick
More informationSCORE. Exam 2. MA 114 Exam 2 Fall 2016
MA 4 Exam Fall 06 Exam Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use
More informationJANE PROFESSOR WW Prob Lib1 Summer 2000
JANE PROFESSOR WW Prob Lib Summer 000 Sample WeBWorK problems. WeBWorK assigmet Series6CompTests due /6/06 at :00 AM..( pt) Test each of the followig series for covergece by either the Compariso Test or
More informationMTH 246 TEST 3 April 4, 2014
MTH 26 TEST April, 20 (PLEASE PRINT YOUR NAME!!) Name:. (6 poits each) Evaluate lim! a for the give sequece fa g. (a) a = 2 2 5 2 5 (b) a = 2 7 2. (6 poits) Fid the sum of the telescopig series p p 2.
More informationTEACHING THE IDEAS BEHIND POWER SERIES. Advanced Placement Specialty Conference. LIN McMULLIN. Presented by
Advaced Placemet Specialty Coferece TEACHING THE IDEAS BEHIND POWER SERIES Preseted by LIN McMULLIN Sequeces ad Series i Precalculus Power Series Itervals of Covergece & Covergece Tests Error Bouds Geometric
More informationSection 5.5. Infinite Series: The Ratio Test
Differece Equatios to Differetial Equatios Sectio 5.5 Ifiite Series: The Ratio Test I the last sectio we saw that we could demostrate the covergece of a series a, where a 0 for all, by showig that a approaches
More informationAdditional Notes on Power Series
Additioal Notes o Power Series Mauela Girotti MATH 37-0 Advaced Calculus of oe variable Cotets Quick recall 2 Abel s Theorem 2 3 Differetiatio ad Itegratio of Power series 4 Quick recall We recall here
More informationSection 7 Fundamentals of Sequences and Series
ectio Fudametals of equeces ad eries. Defiitio ad examples of sequeces A sequece ca be thought of as a ifiite list of umbers. 0, -, -0, -, -0...,,,,,,. (iii),,,,... Defiitio: A sequece is a fuctio which
More informationSolutions to Practice Midterms. Practice Midterm 1
Solutios to Practice Midterms Practice Midterm. a False. Couterexample: a =, b = b False. Couterexample: a =, b = c False. Couterexample: c = Y cos. = cos. + 5 = 0 sice both its exist. + 5 cos π. 5 + 5
More informationSection 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
More informationMath 19B Final. Study Aid. June 6, 2011
Math 9B Fial Study Aid Jue 6, 20 Geeral advise. Get plety of practice. There s a lot of material i this sectio - try to do as may examples ad as much of the homework as possible to get some practice. Just
More informationSection 1.4. Power Series
Sectio.4. Power Series De itio. The fuctio de ed by f (x) (x a) () c 0 + c (x a) + c 2 (x a) 2 + c (x a) + ::: is called a power series cetered at x a with coe ciet sequece f g :The domai of this fuctio
More informationMath 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
More informationMidterm Exam #2. Please staple this cover and honor pledge atop your solutions.
Math 50B Itegral Calculus April, 07 Midterm Exam # Name: Aswer Key David Arold Istructios. (00 poits) This exam is ope otes, ope book. This icludes ay supplemetary texts or olie documets. You are ot allowed
More informationQuiz. Use either the RATIO or ROOT TEST to determine whether the series is convergent or not.
Quiz. Use either the RATIO or ROOT TEST to determie whether the series is coverget or ot. e .6 POWER SERIES Defiitio. A power series i about is a series of the form c 0 c a c a... c a... a 0 c a where
More information1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.
.3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(
More informationPart I: Covers Sequence through Series Comparison Tests
Part I: Covers Sequece through Series Compariso Tests. Give a example of each of the followig: (a) A geometric sequece: (b) A alteratig sequece: (c) A sequece that is bouded, but ot coverget: (d) A sequece
More informationInfinite Series and Improper Integrals
8 Special Fuctios Ifiite Series ad Improper Itegrals Ifiite series are importat i almost all areas of mathematics ad egieerig I additio to umerous other uses, they are used to defie certai fuctios ad to
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationis also known as the general term of the sequence
Lesso : Sequeces ad Series Outlie Objectives: I ca determie whether a sequece has a patter. I ca determie whether a sequece ca be geeralized to fid a formula for the geeral term i the sequece. I ca determie
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationSolutions 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/
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationSolutions to Homework 7
Solutios to Homework 7 Due Wedesday, August 4, 004. Chapter 4.1) 3, 4, 9, 0, 7, 30. Chapter 4.) 4, 9, 10, 11, 1. Chapter 4.1. Solutio to problem 3. The sum has the form a 1 a + a 3 with a k = 1/k. Sice
More informationSolutions 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
More informationIt is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.
Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable
More informationMA131 - Analysis 1. Workbook 3 Sequences II
MA3 - Aalysis Workbook 3 Sequeces II Autum 2004 Cotets 2.8 Coverget Sequeces........................ 2.9 Algebra of Limits......................... 2 2.0 Further Useful Results........................
More informationSCORE. Exam 2. MA 114 Exam 2 Fall 2016
Exam 2 Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use a graphig calculator
More informationx a x a Lecture 2 Series (See Chapter 1 in Boas)
Lecture Series (See Chapter i Boas) A basic ad very powerful (if pedestria, recall we are lazy AD smart) way to solve ay differetial (or itegral) equatio is via a series expasio of the correspodig solutio
More informationAdvanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology
Advaced Aalysis Mi Ya Departmet of Mathematics Hog Kog Uiversity of Sciece ad Techology September 3, 009 Cotets Limit ad Cotiuity 7 Limit of Sequece 8 Defiitio 8 Property 3 3 Ifiity ad Ifiitesimal 8 4
More information1 Lecture 2: Sequence, Series and power series (8/14/2012)
Summer Jump-Start Program for Aalysis, 202 Sog-Yig Li Lecture 2: Sequece, Series ad power series (8/4/202). More o sequeces Example.. Let {x } ad {y } be two bouded sequeces. Show lim sup (x + y ) lim
More informationTHE INTEGRAL TEST AND ESTIMATES OF SUMS
THE INTEGRAL TEST AND ESTIMATES OF SUMS. Itroductio Determiig the exact sum of a series is i geeral ot a easy task. I the case of the geometric series ad the telescoig series it was ossible to fid a simle
More informationSCORE. Exam 2. MA 114 Exam 2 Fall 2017
Exam Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use a graphig calculator
More informationMath 163 REVIEW EXAM 3: SOLUTIONS
Math 63 REVIEW EXAM 3: SOLUTIONS These otes do ot iclude solutios to the Cocept Check o p8. They also do t cotai complete solutios to the True-False problems o those pages. Please go over these problems
More informationA 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
More informationSequences. A Sequence is a list of numbers written in order.
Sequeces A Sequece is a list of umbers writte i order. {a, a 2, a 3,... } The sequece may be ifiite. The th term of the sequece is the th umber o the list. O the list above a = st term, a 2 = 2 d term,
More informationSequences, 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
More informationMH1101 AY1617 Sem 2. Question 1. NOT TESTED THIS TIME
MH AY67 Sem Questio. NOT TESTED THIS TIME ( marks Let R be the regio bouded by the curve y 4x x 3 ad the x axis i the first quadrat (see figure below. Usig the cylidrical shell method, fid the volume of
More informationMath 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
More informationSequences 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 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More information9.3 The INTEGRAL TEST; p-series
Lecture 9.3 & 9.4 Math 0B Nguye of 6 Istructor s Versio 9.3 The INTEGRAL TEST; p-series I this ad the followig sectio, you will study several covergece tests that apply to series with positive terms. Note
More information11.6 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
More informationMATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.
MATH 1080: Calculus of Oe Variable II Fall 2017 Textbook: Sigle Variable Calculus: Early Trascedetals, 7e, by James Stewart Uit 3 Skill Set Importat: Studets should expect test questios that require a
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More information