Fourier Series 1 (Convergence Problems)

Q.

The Fourier series of an odd periodic function, contains only?

1. Sine Terms

2. Cosine Terms

3. Odd Harmonics

4. Even Harmonics

Q. The Fourier cosine series for an even function f(x) given by $f\left(x\right)=a_0+{\sum }_{n=1}^{\infty }{a}_n\cos n\left(x\right)$ The value of the coefficient $a_2$ for the function $f\left(x\right)={\cos }^2\left(x\right)in[0,\pi ]$ is

1. -0.5
2. 0.0
3. 0.5
4. 1.0

Q.

. Let x(t) be a periodic function with period T = 10. The Fourier series coefficients for this series are denoted by $a_k$, that is $x\left(t\right)={\sum }_{k=-\infty }^{\infty }{a}_k{e}^{jk\frac{2\pi }{T}t}$ The same function x(t) can also be considered as a periodic function with period T' = 40. Let $b_k$ be the Fourier series coefficients when period is taken as T'. If ${\sum }_{k=-\infty }^{\infty }|a_k|=16,then{\sum }_{k=-\infty }^{\infty }|b_k|$ is equal to

1. 256
2. 64
3. 16
4. 4

Q. Let g: $[0,\infty )\to [0,\infty )$ be a function defined by g(x) = x - [x], where [x] represents the integer part of x. (That is, it is the largest ineger which is less than or equal to x). The value of the constant term in the Fourier series expansion of g(x) is

1. 0.5

Q.

Fourier series of any periodic signal x(t) can be obtained if
1. ${\int }_0^T|x\left(t\right)|dt<\infty$
2. Finite number of discontinuities within finite interal t
3. Infinite number of discontinuities

Select the correct answer using the codes below.

1. 1, 2 and 3

2. 1 and 3 only

3. 1 and 2 only

4. 2 and 3 only

Q.

The coefficient of $x^3$ in the infinite series expansion of $\frac{2}{(1-x)(2-x)}$, for $\left|x\right|<1$is

1. $\frac{–1}{16}$

2. $\frac{15}{8}$

3. $\frac{–1}{8}$

4. $\frac{15}{16}$