How Did the Author Determine the Fourier Sine Series for x^2?

[iScience]

I'm having trouble understanding a part in my book.

second to last paragraph where it says 4.2 must be the Fourier sine series for x^2, how did the author arrive at that?


http://i.imgur.com/gLLUYXw.jpg

 

[member 428835]

The author was familiar enough with Fourier series to recognize the expression arrived at is the sine expansion of $x^2$. To verify, simply find the Fourier sine series of $x^2$.

There is no reason why most people would recognize this, especially if you're just learning it, so I wouldn't worry about it too much.

 

[LCKurtz]

Something is wrong there. That is not the half range sine for series for $x^2$. The odd periodic extension of $x^2$ is discontinuous so the coefficients could converge no faster than order of $\frac 1 n$. It is in fact the correct sine series for $f(x) = x(\pi -x)$ as is stated.