Sketch a graph of f x in the interval -2pi lt x lt 2pi and

Question

Exercise 1.

Let f (x) be a function of period 2Π such that
                 { 1, -Π < x < 0

f (x) =

                 { 0, 0 < x < Π.

a) Sketch a graph of f (x) in the interval -2Π < x < 2Π

b) Show that the Fourier series for f (x) in the interval -Π < x < Π is

1/2 - 2/Π [sin x + 3 sin 3x + 1/5 sin 5x + ...]

c) By giving an appropriate value to x, show that

Π/4 =  1 - 1/3 + 1/5 -1/7 +.....

Exercise 2.

Let f (x) be a function of period 2Π such that

                 { 0, -Π < x < 0

f (x) =

                  { x, 0 < x < Π.

a) Sketch a graph of f (x) in the interval -3Π < x < 3Π

b) Show that the Fourier series for f (x) in the interval -Π < x < Π is

Π/4 - 2/Π [cos x + 1/3²cos3x + 1/5² cos5x + ...... ] + [sin x - 1/2 sin 2x + 1/3 sin 3x - ...]

c) By giving appropriate values to x, show that

(i) = Π/4 = 1 - 1/3 + 1/5 - 1/7 + ..... and (ii) Π²/8 = 1 + 1/3² + 1/5² +1/7² + .....

Exercise 3.

Let f (x) be a function of period 2Π such that

                 { x, 0 < x < Π

f (x) =

                  {Π, Π < x < 2Π.

a) Sketch a graph of f (x) in the interval -2Π < x < 2Π

b) Show that the Fourier series for f (x) in the interval 0 < x < 2Π is

3Π/4 - 2/Π [cos x + 1/3² cos 3x + 1/5² cos 5x + . . .] - [sin x + 1/2 sin 2x + 1/3 sin 3x + . . .]

c) By giving appropriate values to x, show that

(i) = Π/4 = 1 - 1/3 + 1/5 -1/7 + ...  and (ii) Π²/8 = 1 + 1/3² + 1/5² + 1/7² + ... 

Exercise 4.

Let f (x) be a function of period 2Π such that

             f (x) = x/2 over the interval 0 < x < 2Π.

a) Sketch a graph of f (x) in the interval 0 < x < 4Π

b) Show that the Fourier series for f (x) in the interval 0 < x < 2Π is

Π/2 - [sin x + 1/2 sin 2x + 1/3 sin 3x + . . .]

c) By giving an appropriate value to x, show that

Π/4 = 1 - 1/3 + 1/5 -1/7 + 1/9 - . . .

Exercise 5.

Let f (x) be a function of period 2Π such that

                 { Π - x, 0 < x < Π

f (x) =

                  {0, Π < x < 2Π.

a) Sketch a graph of f(x) in the interval -2Π < x < 2Π

b) Show that the Fourier series for f (x) in the interval 0 < x < 2Π is

Π/4 + 2/Π [ cos x + 1/3² cos 3x + 1/52 cos 5x + ....] + sin x + 1/2 sin 2x + 1/3 sin 3x + 1/4 sin 4x + . . .

c) By giving an appropriate value to x, show that

Π²/8 = 1 + 1/3² + 1/5²  . . .

Exercise 6.

Let f (x) be a function of period 2Π such that

     f (x) = x in the range - Π < x < Π.

a) Sketch a graph of f (x) in the interval -3Π < x < 3Π

b) Show that the Fourier series for f (x) in the interval -Π < x < Π is

2 [sin x - 1/2 sin 2x + 1/3 sin 3x - ..]

c) By giving an appropriate value to x, show that

Π/4 = 1 - 1/3 + 1/5 -1/7 + ....

Exercise 7.

Let f (x) be a function of period 2Π such that

f (x) = x² over the interval - Π < x < Π.

a) Sketch a graph of f (x) in the interval -3Π < x < 3Π

b) Show that the Fourier series for f (x) in the interval -Π < x < Π is

Π²/3 - 4 [cos x - 1/2² cos 2x + 1/3² cos 3x - ...]

c) By giving an appropriate value to x, show that

Π²/6 = 1 + 1/2² + 1/3² + 1/4² + ...