Find a power series representation centered at the origin for the function 1. f(x) 2. f(2) 3. f(x) 4. f(2) 5. f(x) = = = = = f(2) Σ η: Σ n=1 α w Σ n=1 Σ n=1 = α 1 (6 — 2) ². 1 6n+1 1 6n+1 η 6η τη Σ (n+1)=n n=0 τη n-1 η 6n+1 n+1 η 2-1
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- Q#10: Develop f (x) in the Fourier series, if defined as: (i) : f (x)= 3x cos 2x, – aFind a power series representation for the function 1. f(t) 2. f(t) 3. f(t) = = = f(t) i M8 IM8 M8 n=0 n=0 n=0 = 1 5 + 16t² (−1)n 4²nf 2n 5n+1 (−1)n 42n + 2n 5n 42n+2n 5n+10 A projectile fired from the ground follows the trajectory given by the following equation, where is the initial speed, is the angle of projection, g is the acceleration due to gravity, and k is the drag factor caused by air resistance. y =tan(0) + = (tan y = tan (0)x Using the power series representation In(1 + x) = x - x² 2 y =tan(0) + y = (tan(0))x g kvo cos(8) 00 g g kvo cos(8) ) x + 2/2 In (1 k² 9x² 2v² cos²(0) √))x + ²/² ( ²₁ 5-) g Σ n = 1 n = 2 kx Vo cos(8) + x3 3 4 + kgx³ k²gx4 3v³ cos ³ (0) 4v4 cos4(0) -1 < x < 1, verify that the trajectory can be rewritten as the following.Q1. Simplify (cos-isin ) (cos 70+isin 70) (cos 40+ i sin 40)² (cos+ i sin 0)³ using De Movier's Theorem. Q2. Obtain the Power Series solution of the differential equation y"-xy'+y = 0. Q3. Find the Fourier series for the function: f(x)=[x² for 1(a) Find the derivative of the power series ∞ 1 f(z) = −3+ 2πί n=1 (b) Assume a function f is analytic at z = 0, where f(0) derivatives at the origin are given by f(n) (0) in n! n³ z = 0. i Jo 3n (c) Assume a function f is analytic in some neighbourhood of z = √2. Assume further that 1 f(z) (z - √2)n+1 = 5. Assume further that its for n 1. Find its Taylor series about -dz = (n + 5) ³/ for n ≥ 0, where C is a positively oriented circle of radius e centred at z = √2. Find the Taylor series about z = √2 and evaluate the integrals [ f(z) (z − √2)" dz = for all n E N. -3z (d) Find the Laurent series of the function f(z) about z = 0 by using the well-known z5 expression for the exponential function. Where does the series converge?Determine the Fourier series of the function below. f(2)=(6-2), -1dt dt given that x(0) = y(0} = 0. 2. Given f(x)=xif3D1< rland f(x)= f(x+2). (a) Find the trigonometric Fourier series for f (x). Expand the series up to the fourth harmonic. (b) Let x = 1/2 and deduce a series for n/4. Present your final answer using the Sigma notation.P2 Consider the two power series f(x) = x + Σ n=0 g(x) = Σ n=2 (b) Show that f(x) satisfies the equation (3x)2n+1 (2n + 1)! (x + 1)" 3 n ln(n) y" = 9(y-x).2Recommended textbooks for youCalculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSONCalculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage LearningCalculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSONCalculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning