simplifying an expression with even and odd integers

Question

I got this expression for my $b_n$ to a Fourier series: $$b_n=\frac{(2- \pi^2 n^2)\cos(\pi n) -2}{4( \pi n)^3}$$ Now I want to write it in a closed form without the use of $\text{when } n \text{ is even}$ and $\text{when } n \text{ is odd}$. How could this be done?

Answer 1

If n is even then $b_n$ will have the form

$$ b_{2m}=\frac{(2 - 4\pi^2 m^2)\cos(2\pi m) -2}{4( 2\pi m)^3}= -\frac{ 1 }{8\pi m}.$$

I eave it for you to do the odd case. Note that you need $\cos( (2m+1)\pi ) =(-1)^{m+1}$.