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- Q3) a) Evaluate fx-²1₂(3x) dx. b) Expand the following function in a Fourier Bessel series using Bessel functions of the same order as in the indicated boundary condition: ƒ(x)=x², 0SHOW ALL THE STEPS9. If f(x) = {0 when 7/21. Find a power series representation of f(x) = x tan-'x I. (-1)" , ( a) Σ n+1 n=0 (b) E". 2n+1 2n+1 n=0 ( c) Σ (-1)" 2n πο n=0 2n+2 (d) 2 2n+1 (e) 1)"„2n+2 2n+1 n=0 n-08. The Taylor series about x = 3 for a certain function f converges to f(x) for all x in the interval of convergence. The nth derivative off at x = 3 is given by (-1)" n! and ƒ (3) = -3 5" (n+3) f(n) (3)= 1 (c) Show that the third-degree Taylor polynomial approximates f (4) with an error less than 40005. s) Find the Fourier series of the following function and discuss its convergence on the interval -Y, 1< x < 0 f(r) = x. 011. (a) Find the Fourier series of the 27-periodic function given on the interval -<< π by f(x) = sinx. (b) Plot several partial sums to illustrate the convergence of the Fourier series.1). Y Prove that the Fourier series of f(x)=z²+3x², - SIST (-*.*). converges to f uniformly on2. Show that the Fourier series function defined by f(x) below is an even function. Hence determine the Fourier series for the function: f(t)= 1-1, 1+1, when - <1 <0 when 0 <1Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,