Limit of a function vs sequence as $\lim_{n\rightarrow \infty}n\cos (\frac{\pi }{2}+\pi n)$

Question

I know that $\lim_{n\rightarrow \infty}n\cos (\frac{\pi }{2}+\pi n)$ does not exist. However, I am not sure about the sequence $(n\cos (\frac{\pi }{2}+\pi n))_{n=1}^\infty$. By evaluating this sequence, I get indeterminate form $\infty\cdot0$. Intuitevly, I believe that the answer is $0$, because $0$ can be treated as a solid (?). For instance, $\lim_{n\rightarrow \infty}n\cdot0=0$. So, how can I interpret these results?

Answer 1

We have $$ n \cos(n \pi + \frac{\pi}{2})=n(\cos(n\pi)\cos(\frac \pi 2) -\sin(n\pi)\sin(\frac{\pi}{2}))=0\to0. $$