<s>[INST] Simplify $G(t) = \sum_\limits{n=1}^{\infty} (\cos\frac{n\pi}{2}-1) \cos(\frac{nt\pi}{3}) $ Simplify $G(t) = \sum_\limits{n=1}^{\infty} (\cos\frac{n\pi}{2}-1) \cos(\frac{n\pi t}{3}) $ 
I am unsure how to simplify this in the best form, anyone have any ideas?
 [/INST] Here is an alternate representation using the fact that
\begin{equation}
\cos\left(\dfrac{n\pi}{2}\right)-1=\begin{cases}
-1&\text{ for }n=4k-3,\,n=4k-1\\
-2&\text{ for }n=4k-2\\
\phantom{-}0&\text{ for }n=4k\\
\end{cases}
\end{equation}
\begin{eqnarray*}
G(t)&=&-\sum_{k=1}^{\infty}\left[\cos\left(\frac{4k-3}{3}\pi t\right)
+2\cos\left(\frac{4k-2}{3}\pi t\right)
+\left.\cos\left(\frac{4k-1}{3}\pi t\right)\right]\right.
\end{eqnarray*}  
</s>