Question 6 For 0<x<T, Etrol= sin(2n + 1] = . Then the voluo of the series...
(c) A sequence {2n} satisfying 0 < In < 1/n where E(-1)"In diverges.
n=0 4. Using the power series cos(x) = { (-1)",2 (-0<x<0), to find a power (2n)! series for the function f(x) = sin(x) sin(3x) and its interval of convergence. 23 Find the power series representation for the function f(2) and its interval (3x - 2) of convergence. 5. +
3. (20pts.) Find the Fourier series of the function given 0- <x<0 x. 0<x<
Question. Consider () - ( cos(t), sin(t)) for 0 +< 2. Parameterine this curve by are length. Chat
n, fx/<1/2n 5. In the interval (-17, T), O, (x) = jo, x]>1/2n (a) Expand 8, (x) as a Fourier cosine series. (b) Show that your Fourier series agree with a Fourier expansion of d(x) in the limit as n →00.
2) Given that 4 cos[(2n + 1)x] |x| = = = - - nao ,- < x <te. (2n + 1)2 Find the Fourier series of g(x) = -1 1 -1 < x < 0 0 < x <TE 1
Q2: Find the complex Fourier series (show your steps) - T < x <07 f(x) 0 < x < Q1: Find the Fourier transform for (show your steps) - 1<x< 0 Otherwise (хе f(x) = { 0,
Denote the Fourier series of fr-fx, 1<x< 0 f(x) = { 0, 0SX S1 by F(x). Show that E F(x) = - -_ 2500 cos (2mi) + 2m=0 (2m+1) + 500 + 2n=1 + in sin(nx).
Solve fort, 0 < t < 27. 32 sin(t)cos(t) = 12 sin(t)
6. Solve the following boundary value problem: 1 U = 34xx, 0 < x < 1,t> 0; u(0,t) = u(1,t) = 0; u(x,0) = 7 sin nx - sin 31x