I think your bn's are correct except for b2. I getIts hard to explain.
I did this exercise a lot of times, every time I get the same answer.
I can post the whole exercise to help me find what is wrong.
That Bn is kind hard to get right.
That formula doesn't work for n = 2 so you have to do b2 separately. Just directly work out the integralI got this:
Bn = (2*(-1)^n)/(2-n)^2 - (2*(-2)^n)/(2+n)^2
B2 = (2/(2-2)^2) - 1/8
What did you make to get that: (pi^2)/3