Convergence of $\sum_{n=0}^\infty(-1)^n\frac{4^{n-2}(x-2)}{(n-2)!}$ What theorem should I use to show that $$\sum_{n=0}^\infty(-1)^n\frac{4^{n-2}(x-2)}{(n-2)!}$$ is convergent no matter what value $x$ takes?

By "theorem", your teacher probably just means that $(x-2)$ can be moved outside of the series (distributive property over infinite sums), so that its convergence doesn't depend on $x$. Proving the convergence of the series itself would require the series to actually make sense, which it currently doesn't. Though if you nudge the index forward so that all values are defined it will certainly converge, as you can see by applying the ratio test, which may also be what your teacher was referring to.
