differentiating a fourier series

Question

Vocaloid · Answer

this is the fourier series for x^2 http://imgur.com/a/ZzewB I am asked to find a fourier series for x by differentiating the above fourier series. My attempt is to differentiate and then divide by 2 but my answer does not match the answer key

Vocaloid · Answer

my work:$$x = \frac{ 1}{ 2 }\frac{ 4 }{ n^2 }  n  (-1)^{n-1} * n(-\sin(nx))$$

Vocaloid · Answer

$$x = 2(-1)^{n-1}*(-\sin(nx))$$

Vocaloid · Answer

$$x = \frac{ 2(-1)^{n} }{ -1 } * (-\sin(nx))$$

Vocaloid · Answer

$$x = 2(-1)^{n}\sin(nx) = my answer$$

Vocaloid · Answer

however the answer key is $$x = 2(-1)^{n+1}\frac{\sin(nx)}{n}$$

Vocaloid · Answer

(just pretend there are sigma signs in front of each answer, I forgot to draw them)

Vocaloid · Answer

oh wait I figured it out

pandas · Answer

hm...

sillybilly123 · Answer

$(-1)^n = (-1)^{n+1}\cdot \frac{1}{n}$ :)