Study the convergence of sequence $\sin((n\pi)/2)-1$

Question

I have to study the convergence of the sequence $$\sin\left(\frac{n\pi}{2}\right)-1$$

First, I thought of doing it using the squeeze theorem, but the limits are obviously different. Then, I tried by subsequences; I know that the even subsequence goes to $-1$, but I'm stuck with the odd one... It seems to diverge, but I'm not sure of this. Can somebody help me, please?

Answer 1

Let $n=4k+1$, try to evaluate $\sin ( \frac{n \pi}2) -1$. If you get a different value from the even subsequence, then you can conclude that it does not converge.

Remark:

Extra exercise is to examine what happens when $n=4k-1$.