I've got to use the Ratio Test to determine whether this series is convergent or divergent:
$$\sum_{n=1}^\infty \frac{cos(n\pi/3)}{n!}$$
Taking the $\lim \limits_{n \to \infty}|\frac{a_n+1}{a_n}|$ gets me as far as
$$\lim \limits_{n \to \infty}|\frac{cos((n+1)\pi/3)}{cos(n\pi/3)(n+1)}|$$
I'm not sure how to proceed at this point. Unless I'm wrong, $|cos((n)\pi/3)| \lt 1$ for all $n \ge 1$, which should also go for $|cos((n+1)\pi/3)|$, so I can't just take the $\lim$ as ${n \to \infty}$.
Of course $(n+1) \to \infty$ as $n \to \infty$, meaning that as n becomes large, any possible value of the cosine functions should still result in the ratio becoming infinitesimal (since they're both $\le 1$), but I'm not 100% on that, and even so, I'd rather know what the "official" next steps would be in solving the problem.
Best Answer
I'd start with the comparison test: $$\Big|\frac{\cos(\frac{n\pi}{3})}{n!}\Big|\leq \frac{1}{n!} $$ and then you can use the ratio test on the series $\sum_n\frac{1}{n!}$