[Math] Does $\sum_{n=1}^\infty \frac{\cos(n\pi/3)}{n!}$ absolutely converge

Question

Using the Ratio Test, I have to find whether
$$
\sum_{n=1}^\infty \frac{\cos(n\pi/3)}{n!}
$$
converges or diverges. The back of the book says that the sum is absolutely convergent.

My work:

$a_n = \dfrac{\cos(n\pi/3)}{n!}$,
$a_{n+1} = \dfrac{\cos((n+1)\pi/3)}{(n+1)!}$

\begin{align}
&\lim_{n\rightarrow \infty} \left|\frac{a_{n+1}}{a_n}\right| \\[6pt]
\implies&\lim_{n\rightarrow \infty} \left|\frac{\dfrac{\cos((n+1)\pi/3)}{(n+1)!}}{\dfrac{\cos(n\pi/3)}{n!}}\right| \\[12pt]
\implies&\lim_{n\rightarrow \infty} \left|\frac{\cos((n+1)\pi/3) \cdot n!}{\cos(n\pi/3)\cdot(n+1)!}\right| \\[6pt]
\implies&\lim_{n\rightarrow \infty} \left|\frac{\cos((n+1)\pi/3)}{\cos(n\pi/3)\cdot(n+1)}\right| \\
\end{align}

Now this is where I am stuck. I don't know how to find the limit for the $\cos$ terms. I tried using the identity $\cos(x+y) = \cos(x)\cos(y) – \sin(x)\sin(y)$ but it didn't yield anything useful (maybe, I should have tried harder?). I tried looking at this question but it didn't help much.

Any hints would be appreciated.

Thanks for your time!

Answer 1

Use comparison. $${|cos(n\pi x/3)|\over n!} \le {1\over n!}.$$