Question

1) (2 points) Find the Fourier series, which we denote by F(x), of the function f : (-?, 0) ? (0, ?) ? ?, f(x) := x/|x|. Find also the value F(0). 2) (3 points) Find the Fourier sine series, which we denote by F_{sin}(x), of the function f(x) = 4x(? - x) with domain (0, ?). Find also the value of ?_{n=1}^{?} (-1)^n/n^2. 3) (2 points) Find the Fourier cosine series, which we denote by F_{cos}(x), of the function f(x) = { 2 - x/2 if x ? (0, 2), { 0 if x ? (2, ?). 4) (2 points) Find the Fourier series of the function f(x) = |x| on the domain (-1, 1). 5) (2 points) Find the Fourier series of the function f(x) = e^x on the domain (-1, 1). 6) (4 points) Let f be a continuous function with domain (-?, ?), such that f(x + ?) + f(x) = 0 for all x ? (-?, 0). Find the values of a_{2k} and b_{2k} for all k = 1, 2, 3, ··· . (Note: this function f can be quite arbitrary, so you cannot work with any explicit expression of f.)

          1) (2 points) Find the Fourier series, which we denote by F(x), of the function
f : (-?, 0) ? (0, ?) ? ?, f(x) := x/|x|.
Find also the value F(0).
2) (3 points) Find the Fourier sine series, which we denote by F_{sin}(x), of the function f(x) = 4x(? - x)
with domain (0, ?). Find also the value of ?_{n=1}^{?} (-1)^n/n^2.
3) (2 points) Find the Fourier cosine series, which we denote by F_{cos}(x), of the function
f(x) = { 2 - x/2 if x ? (0, 2),
{ 0 if x ? (2, ?).
4) (2 points) Find the Fourier series of the function f(x) = |x| on the domain (-1, 1).
5) (2 points) Find the Fourier series of the function f(x) = e^x on the domain (-1, 1).
6) (4 points) Let f be a continuous function with domain (-?, ?), such that
f(x + ?) + f(x) = 0
for all x ? (-?, 0). Find the values of a_{2k} and b_{2k} for all k = 1, 2, 3, ··· . (Note: this function f can be quite arbitrary, so you cannot work with any explicit expression of f.)

1) (2 points) Find the Fourier series, which we denote by F(x), of the function
f : (-?, 0) ? (0, ?) ? ?, f(x) := x/|x|.
Find also the value F(0).
2) (3 points) Find the Fourier sine series, which we denote by Fsin(x), of the function f(x) = 4x(? - x)
with domain (0, ?). Find also the value of ?n=1^? (-1)^n/n^2.
3) (2 points) Find the Fourier cosine series, which we denote by Fcos(x), of the function
f(x) = 2 - x/2 if x ? (0, 2),
0 if x ? (2, ?).
4) (2 points) Find the Fourier series of the function f(x) = |x| on the domain (-1, 1).
5) (2 points) Find the Fourier series of the function f(x) = e^x on the domain (-1, 1).
6) (4 points) Let f be a continuous function with domain (-?, ?), such that
f(x + ?) + f(x) = 0
for all x ? (-?, 0). Find the values of a2k and b2k for all k = 1, 2, 3, ··· . (Note: this function f can be quite arbitrary, so you cannot work with any explicit expression of f.)

Added by Gloria B.

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