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By integrating the Fourier series equation $$y(x) = \frac{\pi}{2} - \frac{4}{\pi} \sum_{m=0}^{\infty} \frac{cos((2m+1)x)}{(2m+1)^2} $$

term by term from 0 to x, find the function $g(x)$ whose Fourier series is $$g(x) = \frac{4}{\pi} \sum_{m=0}^{\infty} \frac{sin((2m+1)x)}{(2m+1)^3} $$

I don't understand how the negative sign in front of the series in the first equation turns positive, or how the $\frac{\pi}{2}$ disappears/ goes into the series somehow?

user10448599
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  • I think integrating $y(x)$ will give you a function equal to $x \pi/2 - g(x)$, and you can find $g(x)$ from the equality – Jane Doé May 03 '19 at 18:42
  • It doesn't. But you can fiddle with the function you do get to change it into the one you want pretty easily. – MPW May 03 '19 at 18:43

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