Calculate the Sine Fourier series expansion for $\displaystyle f(t) = t^2 $ in $\displaystyle 0 < t < 2.$
I know I need to use $\displaystyle ∑ B_n \sin\left(\frac{nπt}{2}\right)$
I know the answer for $\displaystyle B_n$ is $\displaystyle -\frac{8n}{nπ}$ for even $\displaystyle n$ and $\displaystyle \frac{8}{nπ}-\frac{32}{n^3π^3}$ for odd $\displaystyle n$, but I have no clue how to get there. Thanks for the help.
Here is a general term for $B_n$
$$ B_n = 8\,{\frac { \left( -1 \right) ^{n+1}{n}^{2}{\pi }^{2}+2\, \left( -1 \right) ^{n}-2}{{n}^{3}{\pi }^{3}}}.$$
Now, the Fourier series is given by
$$ t^2 = 8\sum_{n=1}^{\infty}\,{\frac { \left( -1 \right) ^{n+1}{n}^{2}{\pi }^{2}+2\, \left( -1 \right) ^{n}-2}{{n}^{3}{\pi }^{3}}} \sin\left( \frac{n\pi t}{2}\right). $$
You can split the above series for even and odd $n$ if you want.