4. Show that the trigonometric series 8 n=1 sin na Vn converges for all □ € R but is not the Fourier series of a Riemann integrable function on [−1, π].
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- Give the odd extension of f(x) = sin(x/2) defined on the interval (0, T) and find its Fourier-sine series. Where does this series converge as x → T and x → 0 (explain)?Use the substitution t = x - xo to solve the given differential equation. (x + 6)2y" + (x + 6)y' + y = 0 y(x) X,X > -6if Expand f(r) if 0<1. Find the Fourier series of f(x) = x² F(-2)? - x defined for -2 ≤ x ≤ 2 if ƒ (x + 4). What is BONUS: Truncate the series to n = 3, 5, 10. Plot the 3 series in one graph. What is F(−2) for each?using the method of change of parameter, find the general solution of the differential equation.Let f(e) = { , 1 ISI< 2n Find the Fourier series of f (x) over the interval [0,27]6. Find the Fourier series of the function (-Ħ < x < 0) 0, f (x, y) = %3| sin r, (0 < r < n). Use this series to conclude that =1-2 - 2In = 1× (-1)" 4n2-1Find the red letters value.Express f(x) = – x, as a Half Range Fourier sine series over the interval 0Recommended 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,