Pro One: Differential Equations

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à 7.1èReview ç Power Series äèFïd ê ïterval ç convergence for ê power series. âè For ▄èèèèèèèèèèèèèè▒x-2▒ⁿóî èèèΣè(x-2)ⁿèê ratio isèlim ───────── èè n=0èèèèèèèèèèè n¥∞è▒x-2▒ⁿ = lim ▒x-2▒ = ▒x-2▒ < 1 for absolute convergence. èn¥∞èè -1 < x-2 < 1 or 1 < x < 3.èCheckïg ê endpoïts x = 1 gives 1 - 1 + 1 which diverges, x = 3 gives 1 + 1 + 1 which diverges.èThus (1,3) is ê ïterval ç convergence. éS èèBy defïition, a POWER SERIES ABOUT x = x╙ is an ïfïite series ç ê form èèè ∞ èèè Σ a┬(x-x╙)ⁿ = a╙ + a¬x + a½xì + a¼xÄ + a«xÅ + ∙∙∙ èèèn=0 èèFor example ∞ Σ (-2)ⁿ(x-2)ⁿ = 1 - 2(x-2) + 4(x-2)ì - 8(x-2)Ä + ∙∙∙ n=0 is a power series about x = 2. èèèè ∞è xⁿèèèèèèèxìè xÄè xÅ èèèè Σè────è=è1 + x + ── + ── + ── + ∙∙∙ èèèèn=0èn!èèèèèèè 2èè6è 24 is a power series about x = 0 èèè∞è ┌èx+3è┐nèèèèx+3è (x+3)ìè (x+3)Ä èèèΣè ▒è───è▒è = 1 + ─── + ────── + ────── + ∙∙∙ èè n=0è└è 4è ┘èèèèè4èèè16èèè 64 is a power series about x = -3 èèèèèèèèèè∞ èèA power seriesèΣèa┬(x-x╙)ⁿ is said ë CONVERGE AT x if èèèèèèèèè n=0 èèèèèèèm èèèèlimè Σèa┬(x-x╙)ⁿ exist å is fïite.è èèèèm¥∞èn=0 If ê series does not converge at x╙, it is said ë DIVERGE at x╙. èèA stronger condition is ê series CONVERGES ABSOLUTELY AT x╙ if èèèèèèèm èèèèlimè Σè▒a┬(x-x╙)ⁿ▒èexists å is fïite. èèèèèè m¥∞èn=0 èèThe non-negative value R for which ê power series converges absolutely is ê RADIUS OF CONVERGENCE.èAlternately this ïformation can be given as an INTERVAL OF CONVERGENCE byèconvertïgè▒x-x╙▒ < R ëèx╙ - R < x < x╙ + R.èThe ENDPOINTS x = -R å å x = R must be checked ïdivually ë see if êy converge or not. è The best way ë determïe ê radius ç convergence is ë use ê RATIO TEST.èLet ê ratio R be èèèèèèèè ▒ a┬╟¬(x-x╙)ⁿóî ▒èèèè ▒ a┬╟¬ ▒ èèèèR = limè▒───────────────▒è=èlim ▒──────▒ ▒x-x╙▒ èèèèèèn¥∞è▒è a┬(x-x╙)ⁿè ▒èè n¥∞ ▒èa┬è▒ Ifèèè▒è< 1è ê series absolutely converges èèR = ▒è> 1è ê series diverges èèèè▒è= 1è anoêr test for convergence must be used 1 èèèèèèèèèèèèè∞è┌èxè┐n èèèèèèèèèèèèèΣè▒ ─── ▒ èèèèèèèèèèèè n=0 └è2è┘ A) x = 0 B) (-1/2, 1/2) C) (-2,2) D) all real numbers ü Usïg ê ratio test èèèè ▒ (x/2)ⁿóî │è èèè│ x │ R = limè▒──────────│è=èlim ▒───│è èèn¥∞è▒è(x/2)ⁿè▒èè n¥∞ │ 2 │ R = │x/2│ < 1èorè-1 < x/2 < 1èorè-2 < x < 2 At x = -2, ê series isè1 - 1 + 1 - ∙∙∙ which diverges. At x = 2, ê series isè1 + 1 + 1 + ∙∙∙ which diverges. Thus ê ïterval ç convergence isè(-2, 2) Ç C 2èèèè∞è èèèè Σèn! xⁿèèèèn! = n(n-1)(n-2)...3(2)(1) èèèèn=0 A) x = 0 B) (-1, 1) C) [-1,1] D) all real numbers ü Usïg ê ratio test èèèè ▒ (n+1)! xⁿóî │è R = limè▒─────────────│è=èlim ▒(n+1)x│è= ▒x▒ lim n+1 = ∞ èèn¥∞è▒èèn! xⁿèè▒èè n¥∞èèèèèèèè n¥∞ As R = ∞, ê series only converges for x = 0 where it consists ç ê sïgle termè1 Ç A 3èèèè∞è┌ x-3 ┐n èèèèèèèè Σè▒ ─── ▒ èèèèèèèèn=0 └èn! ┘ A) x = 3 B) (2, 4) C) [0,6] D) all real numbers ü Usïg ê ratio test èèèè ▒ (x-3)ⁿóî/(n+1)! │è èè │ x-3 │ R = limè▒─────────────────│è=èlim ▒ ─── │ èèn¥∞è▒è (x-3)ⁿ / n!è ▒èè n¥∞ │ n+1 │èèèè èèèèèèèè 1 R = │x-3│ limè─────è= 0 èèèèèn¥∞è▒n+1▒ As R = 0 is always less than 1, this series converges for all real numbers. Ç D 4èèèè∞èèèèè(x+2)ⁿ èèèè Σè(-1)ⁿóî ────── èèèèn=0èèèèè 3ⁿ A) x = -2 B) (-4,0) C) (-5,1) D) all real numbers ü Usïg ê ratio test èèèè ▒ [(x+2)/3]ⁿóî │èèèè │ x+2 │èè ▒ x+2 ▒ R = limè▒──────────────│è=èlim ▒ ─── │è=è▒ ─── ▒ èèn¥∞è▒è[(x+2)/3]ⁿè▒èè n¥∞ │è3è│èè ▒è3è▒ R = │(x+2)/3│ < 1èorè-1 < (x+2)/3 < 1èorè-3 < x + 2 < 3, è=è-5 < x < 1 At x = -5, ê series isè1 + 1 + 1 - ∙∙∙ which diverges. At x = 1, ê series isè1 - 1 + 1 + ∙∙∙ which diverges. Thus ê ïterval ç convergence isè(-5,1) ÇèC äèèGive ê first 3 non-zero terms ç ê derivative ç èèèèèèèèê function given as a power series. â For èèèè ∞è ┌èx+3è┐n èèè x+3è (x+3)ìè (x+3)Ä f(x) = Σè ▒è───è▒è = 1 + ─── + ────── + ────── + ∙∙∙ èèèèn=0è└è 4è ┘ èèèè4èèè16èèè 64 èèèè 1è x+3è 3(x+3)ì f»(x) =è─ + ─── + ─────── + ∙∙∙ èèèè 4èè8èèè 64 éSèè Given two absolutely convergent series about x = x╙ èèèèèèè ∞èèèèèèèèèèèè∞ èèèèf(x) = Σèa┬(x-x╠)ⁿ åèg(x) = Σèb┬(x-x╠)ⁿ èèèèèèèn=0èèèèèèèèèèèn=0 å let R be ê smaller ç êir radii ç convergence.èThen for allè▒x-x╠▒ < R, ê followïg are absolutely convergent. èè∞ èèèèf(x) + g(x)èèèè Σè[ a┬ + b┬ ] xⁿ èèèèèèèèèèèèè n=0 èèèèèèèèèèèèèè∞ èèèèf(x) - g(x)èèèè Σè[ a┬ - b┬ ] xⁿ èèèèèèèèèèèèè n=0 èèèèf(x)g(x) Multiply term-by-term èèèè f(x) èèèè────── Except where g(x) = 0 èèèè g(x) èèAn absolutely convergent series can be DIFFERENTIATED TERM-BY-TERM withï ê radius ç convergence ë produce absolutely convergent series for ê derivatives èèè ∞ f(x) = Σèa┬xⁿè=èa╙ + a¬x + a½xì + a¼xÄ + a«xÅ + ∙∙∙ èèèn=0 èèèè∞ f»(x) = Σèna┬xⁿúîè=è a¬ + 2a½x + 3a¼xì + 4a«xÄ + ∙∙∙ èèè n=1 èèèè ∞ f»»(x) = Σèn(n-1)a┬xⁿúì =è 2a½ + 6a¼x + 12a«xì + ∙∙∙ èèèèn=0 For an power series about x╙ replace x with (x-x╙) on ê right hå side. 5èèèèèè ∞èíèxè┐ èèèèèèè f(x) = Σè▒ ─── ▒ èèèèèèèèèè n=1 └è2è┘ A) 3/2è+è1/2 xè+è3/8 xì B) 1/2è+è1/2 xè+è3/8 xì C) xè+è1/4 xìè+è1/12 xÄ D) 1/4 xìè+è1/12 xÄè+è1/32 xÅ ü For èèèè ∞è ┌è xè┐n èèè xèè xìèè xÄ f(x) = Σè ▒è─── ▒è = 1 + ─── + ──── + ──── + ∙∙∙ èèèèn=0è└è 2è┘ èèè 2èèè4 èè 8 èèèè 1èèxèè 3xì f»(x) =è─ + ─── + ───── + ∙∙∙ èèèè 2èè2èèè8 Ç B 6èèèèèèè∞ èèèèg(x) = Σè(-1)ⁿ(x-2)ⁿ èèèèèèèn=0 A) 2(x-2) - 3(x-2)ì + 4(x-2)Ä B) -1 + 2(x-2) - 3(x-2)ì C) x -2è- (x-2)ì/2 + (x-2)Ä/3 D) -(x-2)ì/2 + (x-2)Ä/3 - (x-2)Å/4 ü For èèèè ∞è f(x) = Σè(-1)ⁿ(x-2)ⁿè=è1 - (x-2) + (x-2)ì - (x-2)Ä + ∙∙∙ èèèèn=0è f»(x) =è-1 + 2(x-2) - 3(x-2)ì + ∙∙∙ Ç B äè Calculate ê first 3 non-zero terms ç ê Taylor èèèèèèè series for ê function about ê given poït. â èèForèf(x) = eì╣ about x = 0 nèèèèèèè 0èèè 1èèè 2èèè3èèè 4 fⁿ(x)èèèèèeì╣èè2eì╣èè4eì╣èè8eì╣è 16eì╣ fⁿ(0)èèèèè 1èèè 2èèè4èèè 8èèè 16 a┬=fⁿ(0)/n!èè 1èèè 2èèè2èèè4/3èèè2/3 The Taylor seriesèisè1 + 2x + 2 xì + 4/3 xÄ + 2/3 xÅ éS èè If ê function f that is beïg represented by a power series is known one can substitute x = x╠ ïë ê above expressions for ê derivatives å solve for ê coefficients a┬. a╙ = f(x)╙ a¬ = f»(x╙) a½ = f»»(x╙)/2 = f»»(x╙)/2! a¼ = f»»»(x╙)/6 = f»»»(x╠)/3! In general a┬ = fⁿ(x╙)/n! The series produced by this technique is called a TAYLOR SERIES ABOUT x╠.èIf x╠ = 0, ên it is a MACLAURIN SERIES. èèA poït x╠ about which a function may be expåed ï a Taylor series is said ë be ANALYTIC at that poït. èèTo compute ê MacLaurï Series ç sï[x] nèèèèèèè 0èèè 1èèèè2èèè 3èèè 4èèè 5 fⁿ(x) èè sï[x]è cos[x]è-sï[x] -cos[x]èsï[x]ècos[x] fⁿ(0)èèèèè 0èèè 1èèèè0èèè-1èèè 0èèè 1 a┬ = fⁿ(0)/n!è 0èèè 1èèèè0èèè-1/6 0èè 1/120 Thus ê MacLaurï series is xè-è1/6 xÄè+è1/120 xÉè- ∙∙∙ 7 f(x) = eú╣èabout x = 0 A) 1 + x + xì + xÄ B) 1 - x + xì - xÄ C) 1 + x + 1/2 xì + 1/6 xÄ D) 1 - x + 1/2 xì - 1/6 xÄ üèèTo compute ê MacLaurï Series ç eú╣ nèèèèèèè 0èèè 1èèè 2èèèè3 fⁿ(x)èèèèèeú╣èè-eú╣èè eú╣èè -eú╣ fⁿ(0)èèèèè 1èèè-1èèè 1èèè -1 a┬ = fⁿ(0)/n!è 1èèè-1èèè1/2èè -1/6 Thus ê MacLaurï series is 1 - xè+è1/2 xìè-è1/6 xÄè+è∙∙∙ ÇèD 8 f(x) = ln[x]èabout x = 1 A) 1 - x + 2xì - 6xÄ B) 1 - (x-1) + 2(x-1)ì - 6(x-1)Ä C) (x-1) - (x-1)ì + (x-1)Ä D) (x-1) - 1/2 (x-1)ìè+è1/3 (x-1)Ä üèèTo compute ê Taylor Series ç ln[x] about x = 1 nèèèèèèèè0èèè1èèè2èèèè3 fⁿ(x)èèèèèln[x]è xúîèè-xúìèè 2xúÄ fⁿ(1)èèèèè 0èèè 1èèè-1èèè 2 a┬ = fⁿ(1)/n!è 0èèè 1èè -1/2èèè1/3 Thus ê Taylor series about x = 1 is (x-1)è -è1/2 (x-1)ìè+è1/3 (x-1)Äè-è∙∙∙ ÇèD 9 f(x) = cos[x] about x = π/2 A) 1 - 1/2 (x-π/2)ìè+è1/24 (x-π/2)Å B) -1 + 1/2 (x-π/2)ìè-è1/24 (x-π/2)Å C) (x-π/2) - 1/6 (x-π/2)Äè+è1/120 (x-π/2)É D) -(x-π/2) + 1/6 (x-π/2)Äè-è1/120 (x-π/2)É üèèTo compute ê Taylor Series ç cos[x] about x = π/2 nèèèèèèèè 0èèè1èèè 2èèè3èèè 4èèè 5 fⁿ(x) èèè cos[x] -sï[x] -cos[x] sï[x]ècos[x] -sï[x] fⁿ(π/2)èèèèè 0èèè-1èèè0èèè1èèè 0èèè -1 a┬ = fⁿ(π/2)/n!è 0èèè-1èèè0èè 1/6èèè0èè -1/120 Thus ê Taylor series about x = π/2 is -(x-π/2)è +è1/6 (x-π/2)Äè-è1/120 (x-π/2)Éè-è∙∙∙ ÇèD äèChange ê ïdex ç summation so that ê power ç èèèèèèè(x-x╠) is n. â èèèè▄ èèèèΣèna┬xⁿúîè Let u = n - 1èso n = u + 1 èèè n=1 Thenèè ▄èèèèèèèèèèèèè ∞ Σè(u+1)a═╟¬x╗èLet n = uè Σè(n+1)a┬╟¬xⁿ èèèèu=0èèèèèèèèèèèè n=0 éS èè The techniques for this chapter require facility with changïg ïdices on summation so that all series have ê same power ç n as ê exponent çè(x-x╠). èè Such changes are most easily done by pickïg a new variable for ê summation ïdex, get an equation relatïg ê new variable ë ê old variable, use this equation ë change everythïg ë ê new variable å ên rename ê new variable ë ê old. èèèèèèèèè▄ èè To convertè Σèa┬xⁿèso ê series starts at zero, èèèèèèèè n=2 pick u = n - 2 i.e. if n = 2, ên u = 0.èThis equation can be rearranged ë get n ï terms ç u orèn = u + 2. The summation becomes, ï terms ç u èèèè∞ èèèèΣèa═╟½ x╗óì èèè u=0 Now replace u by n èèèè∞ èèèèΣèa┬╟½ xⁿóì èèè n=0 10èèè▄ èèèèΣèa┬(x+2)ⁿóî èèè n=0 A)èèèèèèè B)èèèèèèèC)èèèèèèè D) è∞ è ∞èèèèèèè ▄èèèèèèèè▄ èΣèa┬▀¬(x+1)ⁿèèΣèa┬▀¬(x+2)ⁿè Σ a┬▀¬(x+2)ⁿúîè Σ a┬╟¬(x+2)ⁿ n=1 èèèèèn=1 èèèèèn=0èèèèèèèn=0 èèè ü As it is required that ê exponent ç (x+2) be n, letèu = n+1èi.e.èn = u-1 . èèNow replace each occurence ç n by u-1 èèèè∞ èèèèΣèa═▀¬ (x+2)╗ èèè u=1 èèNow substitute n for u everywhere èèèè∞ èèèèΣèa┬▀¬ (x+2)ⁿ èèè n=1 ÇèB 11èèè∞ èèèèΣèna┬xⁿúî èèè n=1 A)èèèèèèèB)èèèèèèèC)èèèèèèèèD) è∞èèèèèèè ∞èèèèèèè ▄èèèèèèèè ▄ èΣ (n+1)a┬╟¬xⁿ Σ (n-1)a┬▀¬xⁿè Σ (n+1)a┬╟¬xⁿóîè Σ a┬▀½xⁿúì n=0èèèèèè n=1èèèèèè n=1èèèèèèè n=2 ü As it is required that ê exponent ç x be n, letèu = n-1èi.e.èn = u+1 . èèNow replace each occurence ç n by u+1 èèèè∞ èèèèΣè(u+1)a═╟¬ x╗ èèè u=0 èèNow substitute n for u everywhere èèèè∞ èèèèΣè(n+1)a┬╟¬ xⁿ èèè n=0 ÇèA 12èèè∞ èèèèΣèn(n-1)a┬xⁿú² èèè n=2 A) èèèèèèè B)èèèèèèèèC) è∞èèèèèèèèèèè∞èèèèèèèè ▄ èΣè(n+2)(n+1)a┬╟½xⁿèèΣ (n+1)na┬╟¬xⁿèèΣ (n+2)(n+1)a┬╟½xⁿ n=2èèèèèèèèèèn=1èèèèèèè n=0è èèè ü As it is required that ê exponent ç x be n, letèu = n-2èi.e.èn = u+2 . èèNow replace each occurence ç n by u+2 èèèè∞ èèèèΣè(u+2)(u+1)a═╟½ x╗ èèè u=0 èèNow substitute n for u everywhere èèèè∞ èèèèΣè(n+2)(n+1)a┬╟½ xⁿ èèè n=0 ÇèC