1. ## Absolute Convergence Problem

1.) Determine if the series is absolutely convergent, conditionally convergent or divergent: $\displaystyle \sum_{n=1}^{\infty} \frac{cos(n\pi/3)}{n!}$

I think we use the Ratio Test for this one but can someone please clarify the steps I need to take to finish this problem? Thanks!

2. Here is a strong hint.
$\displaystyle \left| {\frac{{\cos \left( {\frac{{2n\pi }} {3}} \right)}} {{n!}}} \right| \leqslant \frac{1} {{n!}}$

3. Originally Posted by amp10388
1.) Determine if the series is absolutely convergent, conditionally convergent or divergent: $\displaystyle \sum_{n=1}^{\infty} \frac{cos(n\pi/3)}{n!}$

I think we use the Ratio Test for this one but can someone please clarify the steps I need to take to finish this problem? Thanks!
Plato obviously used the right method...but just to reinforce it if you used the ratio test you would have $\displaystyle \lim_{n \to {\infty}}\bigg|\frac{cos(\pi(n+1))}{(n+1)\cdot{n!} }\cdot\frac{n!}{cos(\pi{n}})\bigg|<1$

then by simplification we would have $\displaystyle \lim_{n \to {\infty}}\bigg|\frac{cos(\pi(n+1))}{(n+1)cos(\pi{n })}\bigg|<1$...now how are you going to simplify that? this is why the direct comparison test is a better choice...hope this helps!