lim cos[pie]n/2
n-[inf]
does converges? If it does what is the limit?
Are you doing $\displaystyle \lim_{n\to\infty}\cos\frac{\pi n}2?$
What is the cosine of a multiple of $\displaystyle \frac\pi2?$ Look at the unit circle: if $\displaystyle n$ is odd, you get 0, and for even $\displaystyle n$ the value alternates between 1 and -1.
Is the sequence approaching a single finite value?