Fourier Coefficients and Final Expected Value
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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.
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This solution is comprised of a detailed explanation to solve Fourier Coefficients problem.
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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 ...
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