Question 4 (30 points) For the given periodic function
f(x) = 3π 3π f(x) FS, f(x) = (π)/(2) - (4)/(π) ∑_(n=1)^(∞) (3,5dots)((1)/(n^(2))cos(nx))(π^(2))/(8) = 1 + (1)/(3^(2)) + (1)/(5^(2)) + (1)/(7^(2)) + cdots and (π^(2))/(6) = 1 + (1)/(2^(2)) + (1)/(3^(2)) + (1)/(4^(2)) + (1)/(5^(2)) + cdots xf(x)f(x) = |x|, -π
(a). (7 points) Sketch f(x) on the interval -3π 3π.
(b). (10 points) Show that the Fourier series of f(x) satisfies
FS, f(x) = (π)/(2) - (4)/(π) ∑_(n=1)^(∞) (3,5dots)((1)/(n^(2))cos(nx))
(c). (10 points) Based on the result of 4(b), show that
(π^(2))/(8) = 1 + (1)/(3^(2)) + (1)/(5^(2)) + (1)/(7^(2)) + cdots and (π^(2))/(6) = 1 + (1)/(2^(2)) + (1)/(3^(2)) + (1)/(4^(2)) + (1)/(5^(2)) + cdots
(d). (3 points) Does there exist some values of x, for which the series fails to converge to f(x)? If
yes, to what values does it converge at those points? If not, justify your result.
Question 4 (30 points) For the given periodic function
f(x) = |x| > x > -
f(x + 2T) = f(x)
(a). (7 points) Sketch f(x) on the interval [-3 3]. (b). (10 points) Show that the Fourier series of f(x) satisfies
FSf(x) = r 2
(c). (10 points) Based on the result of 4(b), show that TT2 1 8 52 TT2 and 6 32 42 (d). (3 points) Does there exist some values of x, for which the series fails to converge to f(x)? If yes, to what values does it converge at those points? If not, justify your result.