Fourier series convergence question

1. Sep 9, 2011

JamesGoh

1. The problem statement, all variables and given/known data

f(x) = 5, -pi <= x <= 0
f(x) = 3, 0 < x <= pi

f(x) is the function of interest

Find the x-points where F(x) fails to converge
to f(x)

2. Relevant equations

F(x) = f(x) if f is continuous at $x\in(-L,L)$
F(x) = 0.5[ f(x-) + f(x+) ] if f is discontinous at $x\in(-L,L)$

F(x) is the fourier series of f(x)

3. The attempt at a solution

Would the Fourier series, F(x) fail to converge
at +pi and -pi ?

My reasoning is as follows

- At both -pi and +pi, we have the start and end value of f(x)

- By definition F(x) = (1/2)f(x-) + (1/2)f(x+)

- Lets take x = -pi, which means f(x) =5 and f(-x) = 3

- F(-pi) = (1/2)(5) + (1/2)(3) = 8/2 = 4 which is not equal to f(-pi)=5

- From the latter, we can conclude that F(-pi) fails to converge to
the value given from f(-pi) since their not equal to each other

Just want to check if my reasoning is ok since no answer was provided to this question

thanks

2. Sep 9, 2011

micromass

Staff Emeritus
Seems ok...

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