Which sign does $\lim_{n\to\infty}(-2)^n$ have?

Question

How can I express $\lim_{n\to\infty}(-2)^n$ using $\infty$? Which sign does it have, plus or minus?

Answer 1

Neither. $\lim_{n \to \infty}(-2)^n$ does not exist.

Answer 2

In some contexts it makes sense to have just a single $\infty$ that is approached if you go in either direction, rather than the two separate objects $\pm\infty$. Whenever that is what "$\infty$" is taken to mean, then your sequence approaches $\infty$. But it does not approach either $+\infty$ or $-\infty$, since its sign alternatves.