Determine the interval of convergence

$\displaystyle \sum \frac{-1^k(x+\frac{\pi}{2})^{2k+1}}{2k+1}$

I used the ratio test and got it down to

$\displaystyle \frac{1}{2}\lim= -(x+\frac{\pi}{2})^3$

but I think I messed up somewhere or maybe I shouldn't use the ratio test and maybe try something different.