Fourier series convergence test

  • Thread starter JamesGoh
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Homework Statement



A function f(x) is given as follows

f(x) = 0, , -pi <= x <= pi/2
f(x) = x -pi/2 , pi/2 < x <= pi

determine if it's fourier series (given below)

[itex]F(x)=\pi/16 + (1/\pi)\sum=[ (1/n^{2})(cos(n\pi) - cos(n\pi/2))cos(nx)
- (1/(2n^{2}))( n\pi cos(n\pi) + 2sin(n\pi/2) )sin(nx) ][/itex]

for n= 1 to infinity


converges to it for the case of x = pi/2

Homework Equations



F(x) = 0.5*[f(x+) + f(x-) ]


The Attempt at a Solution



see pdf attachment
 

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Answers and Replies

  • #2
vela
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Your work is kind of hard to follow, but it looks like you incorrectly got rid of the cosine terms. You have

[tex]F(x)=\frac{\pi}{16} + \frac{1}{\pi}\sum_{n=1}^\infty \frac{\cos n\pi -\cos \frac{n\pi}{2}}{n^2} \cos nx - \frac{1}{\pi}\sum_{n=1}^\infty \frac{n\pi \cos n\pi + 2\sin \frac{n\pi}{2}}{2n^2}\sin nx [/tex]

You can simplify it a bit by noting that [itex]\cos n\pi = (-1)^n[/itex]. Next, when [itex]x=\pi/2[/itex], the values of cos nx and sin nx depend on whether n is odd or even. Use those facts to get the series for [itex]F(\pi/2)[/itex].
 
Last edited:

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