Determine the power series expansion (in sigma notation) of the function f (r) = sin a at a = 0 Note: The limit of sigma notation in the choices are from n=0 to n=infinity. (-1)2n Σ (n+1)! Σ (-1)" 2n+1 (2n+1)! Σ (-1)m (2n)! 2+1 (-1)" 2n (2n+1)
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- C: Question 16 Write the Taylor series of the function f(x) = x cos(4x) about x = 0. %3D (-1)*r2 k+4 42 k (2 k)! a) OE k=0 5(-1)*4²kx2 k+4 (2 k + 4)! 2k2 k+4 b) k=0 k+42 c) (2 k)! k=0 54(-1)*x2k+4 (2 k)! d) O2 k=0 (-1)*4²kx2 k+4 e) O (2 k)! k=0 (-1)*4² k+1x2 k+5 (2 k+ 1)! k=0What functions in the range - < x < π do the following Fourier series represent? (i) Σ n=0 (ii) Σ1(−1)n+1 cos na n sin(2n+1)x 2n+11. Determine the FOURIER COSINE SERIES of the following STEP FUNCTION: (-1,0 < x < 2 f(x) = } 0,2 < x < 4 (4x,4 < x < 6
- SET 1 Find the Fourier Series expansion of the following function and draw three periods of the graph of f(x). x + if - < x < n f(x) = { T if 0 < x≤ T Where f(x) has the period of 2π. A BI 7 UNH IIIFind the Fourier series expansion of the function. -n < x < 0 f(x) = < x < TSolve the following differential equation: dy x + y? dx 2хy
- QUESTION 6 (a) The function et is to be approximated by a fifth-order polynomial over the interval [-1, +1]. Why is a Chebyshev series a better choice than a Taylor (or Maclauring) expansion? (b) Given the power series and the Chebyshev polynomials f(x)=1-x-2x² - 4x4 To (x) T₁ (x) T₂ (x) T3 (x) = 1 = X = 2x² - 1 = 4x³ - 3x T4(x) = 8x48x² +1, economize the power series f(x) twice. (c) Find the Padé approximation R3 (x), with numerator of degree 2 and denominator of degree 1, to the function f(x) = x² + x³.Expand the following periodic function into a Fourier series. for -2 < <-T < 0 -2 A f(x) = 1 0 for T-X for X-T for k <-T 0 < ㅠ XXExample 8.6 Find the Laurent series expansion for f(:) - 3)(2 – 1) (a) on {2:1< |= < 3}. (b) on a punctured disc centred at z 1. Where this series valid?The real form of the Fourier sine series for the following function: (-1), - T- A Fourier Series a.) Determine the Fourier series expansion for the periodic extension of (x, ß) cosh(ßx) for -n < x < T, where B is a fixed positive constant. b.) Then use this to compute exact values for the sums (-1)" + B2 1 Σ Σ and n? n=1 + B2 n=1 as functions of ß for 0 < B. Hint: Try evaulating f(x,ß) at different values of x.Find the Fourier series of following functions 1. Find a Fourier series to represent, f(x) = a – x for 0Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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