Question

differentiating a fourier series

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

oh wait I figured it out

Answer 9

hm...

Answer 10

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