Find the Fourier series expansion of In(3 – x) around 1. х — 1 (х — 1)2 (х- 1)3 a. In(2) 8 24 b. х — 1. (х — 1)? (х — 1)3 + In(2) + 8 24 х — 1 (x – 1)² (x – 1)³ In(2) - 2 3 4 d. (х — 1)3 +. 4 (x – 1)2 X - In(2) + 2 + 3
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- Find the Fourier series expansion of e3-2x² around -1. O a. 8e e + 4e(x + 1) + 6e(x + 1)² + (x + 1)³ + ... Ob. e + 4e(x + 1) + 6e(x + 1)² + 8e(x + 1)³ + ... Oc. 8e e – 4e(x + 1) + 6e(x + 1)² (x + 1)³ + ... 3 Od. e – 4e(x + 1) + 6e(x + 1)² – 8e(x + 1)³ + ...c) For a periodic function y(t) with period T, its Fourier series may be represented in complex form as y(t) = Σ n=1∞ where w = 2π/T and all cn are complex constants defined for n integer. derive a formula for Cn. = i. Show that if y(t) is a real function satisfying y where the bar denotes the complex conjugate. ii. Using the fact that 2πT Cneinwt ei(n-m)t dt S 2π { = 0 y, then c_n = if m= n if m‡n Cn for all integers n, = CnThe fourier series for a function f(x) = 1+π2/3 + 4(−cos(x)+(1/4)cos(2x)−(1/9)cos(3x)+(1/16)cos(4x)−(1/25)cos(5x) +...) + 4(−sin(x)+(1/2)sin(2x)−(1/3)sin(3x)+(1/4)sin(4x)−(1/5)sin(5x)+...). Find the value for b33.
- (a) Prove that n + 2n is divisible by 3 for each n EN. (b) Let z = sin 1 (in radians) and let y = v2. Prove that the following statement is false: (z+ y € Q) A (1- y E Q). (c) Let ai = 1 and let a2 = n. For n > 3, we define recursively an = an-2 – an-1. Is it true that a2021 is rational? Prove that your answer is correct.Find the Fourier series expansion of In(3 – x) around 1. Oa. x - 1 (x – 1)² (x – 1)³ In(2) + ... 8 24 Ob. x –1 (x – 1)² (x – 1)³ In(2) | 8 24 x- 1 In(2) – (x – 1)2 (x – 1)³ | 4 d. X-1 (x – 1)2 (x – 1)³ In(2) + 2 ... 4 3. 3. 2] 2. 2.1. Find the Fourier series expansion of the following periodic function: -2 < t <0 0Consider the Fourier expansion f(x) = = ∞ where Ck (−1)²k+1 2 k Describe the periodic function f(x). k=1 C₁ = - Verify that the first Fourier coefficient satisfies CTT [ ㅠ Ck sin(kx), xsin (x) dx.Find the trigonometric Fourier series for the function f(x) : [−1, 1] → R given by the expression: f(x) = O O [0 if x = [-1,0] 11+xifre (0, 1] FS(x) = ³ + x f +Σ, ( n=1 FS(r) = ² + x1 ( n=1 8 FS(x) = ² + 1 FS (x) = ²/1 + Σ1 // n=1 (-1)"+1 n²π² (-1)^-1 NT (-1)"-1 n² π² cos(nлx) - - cos(nлx) + 1+2(-1)" na 1+2(-1)" nn 1-2(-1)" cos(nπx) + ¹-2-¹) sin(nπx) sin(nTr)). COS sin(nπx)) a)). 1-2(-1)" NT sin(nπx) -cos(nπx) + -¹ sin(nra)). (-1)" —1 7² 7²Find the Fourier Series of the given periodic function. f(0) = {-1, if -n≤t<0 if ostSolve c and d, using the table might help1. Find the Fourier series expansion of the following periodic function: -2 < t <0 03. Find the fourier series expansion of the periodic function f(r) = e², -a < r < 7, 1 1 f(1+ 2n) = f(r). Hence obtain the sum of the series 1 1+22 1+32'1+ 4? (-1)" +.... 1+n?SEE MORE QUESTIONSRecommended 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