# Fourier Coefficients and Final Expected Value

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.

© BrainMass Inc. brainmass.com June 3, 2020, 5:54 pm ad1c9bdddfhttps://brainmass.com/math/fourier-analysis/fourier-coefficients-final-expected-value-41865

#### Solution Preview

Solution:

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.