Here, we have to find f(t) from the given value of Cn. I am not able to arrive at f(t)={3/[5-4cos(pit+pi/20)} despite many attempts. Please show me how to arrive at the final expected f(t) value.

There should be a mistake concerning the coefficients.
I think (in fact, I'm sure) that the correct coefficients are:
C_n = (1/2)^n*e^(i*n*(pi)/20) for n >= 0 and (1)
C_n = 2^n*e^(i*n*(pi)/20) for n < 0

The explanation is that series needs to be convergent and, if n < 0, the modulus of coefficients on the left side of series where n < 0 (as they are given in problem) would be (1/2)^(-n) = 2^n, that is greater than unity, so that series would be divergent.

Thus, let's consider (1) as true and let's proceed to compute the sum of series:
s = f(t) = 1 + sum (from 1 to inf) of (1/2)^n*e^(i*n*(pi)/20)*e^(i*n*(pi)*t)+
+ sum (from -inf to -1) of 2^n*e^(i*n*(pi)/20)*e^(i*n*(pi)*t) (2)
(see final remark)
--> s = 1 ...

Solution Summary

This solution is comprised of a detailed explanation to solve Fourier Coefficients problem.

There is an attached file with further information regarding the problem.
Find by inspection the first seven Fouriercoefficients {a0, a1, b1, a2, b2, a3, b3} of the function:
f(x) = 14-cos(Pi*x/10) + 3sin(Pi*x/10) + 0.5cos(Pi*x/5) + 5sin(3*Pi*x/10)

Compute the Fourier series coefficient of the periodic function shown in the figure(view attachment) and defined as:
X(t) = A 0 <= t <= T/3
0 T/3 <= t < T

3. Solve the wave equation,
∂2u/∂t2 = c2(∂2u/∂x) -∞ < x < ∞
With initial conditions, u(x,0) = (1/x2+1)sin(x), and ∂u/∂t(x,0) = x/(x2+1)
4. Suppose that f is a 2п-periodic differentiable function with Fouier coefficients a0, an and bn. Consider the Fourier coeffici

We use the Fourier expansions of certain poynomial functions to compute the sum of some useful numerical series.
The formulas are quite general and give, at the end, the Fourier expansion of every polynomial function.
By the way, these formulas can be also used for a numerical approximation of pi=3.14....

1. Find the Fourier sine series of f(x)=1, 0L/2
Please see the attachment to view the questions with correct mathematical notation (and also phrased slightly differe

Fouriercoefficients / b1, b2, b3, b4, b5... b11.
--------------------------------------------------------------------------------
I have an output of an electronic device (full wave rectifier) that gives a sine wave with the negative part transposed symmetric to xx so that the function is always positive. I have to find the f

Please see attachment.
1. What is the Fourier Transform for the convolution of sin(2t)*cos(2t).
2. Compute the inverse Fourier transform for X(w)= sin^2*3w
3. A continuous time signal x(t) has the Fourier transform
X(w) = 1/jw+b where b is a constant. Determine the Fourier transform for v(t) = x*(5t-4)