# Fourier Series of a Periodic Fuction : Convergence

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Work out the Fourier series of f, given over one period as follows.

At which values of x does the series fail to converge to f(x)? To what values does it converge at those points?

(h) |cos x| for all x

(k) x on 0<x<1, 1<x<2.

See the attached file.

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Convergence of Fourier Series of Periodic Fuctions is investigated. The solution is detailed and well presented.

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Please see the attached file for the complete solution.

Solution. By a theorem which states, if y=f(x) is a periodic function and y=f(x) is continuous at x=a, then its Fouries series converges to x=a; if y=f(x) is NOT continuous at x=b, then its Fouries series fails to converge to x=b. So,

(a) Its Fourier series converges to all points in and ...

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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