Using the root test when the limit does not exist

The root test can be used without the sequence having a limit. Precisely,

if there exist $N$ and $c<1$ with $\sqrt[n]{|a_n|}\le c$ for all $n>N$, then the series $\sum_{n=0}^\infty a_n$ is absolutely convergent.

Indeed, in this case one can directly compare the series with a convergent geometric series. When $\lim_{n\to\infty}\sqrt[n]{|a_n|}=l$ exists and is $<1$, then the above criterion applies, because we can take $c=(l+1)/2$.

If you had used the “extended criterion” rather than stating that $\lim_{n\to\infty}\lvert\frac{\cos n}{2}\rvert\le \frac{1}{2}$, you would be right.

We have that

$$ \left|\left(\frac{\cos n}{2}\right)^n\right|\le \frac1{2^n}$$

and $\sum \frac1{2^n}$ is a convergent geometric series, we don't need root test here.

Anyway we can also apply root test to the original series in the general form by limsup definition

$$\limsup_{n\rightarrow\infty}\sqrt[n]{\left|\left(\frac{\cos n}{2}\right)^n\right|}=L\le \frac12$$

and conclude that the series converges.