Infinity Series problem with cos $$\sum_{n=1}^\infty\frac{\cos(3n+1)}{3^n},\quad (a_n)=\frac{\cos(3n+1)}{3^n}$$
I have to find out if it converges/diverges or not
My starting approach was that since $\cos(3n+1)<1 \implies \cos(3n+1)/3^n < 1/3^n$
then since $\sum_{n=1}^\infty1/3^n$ converges my sum will converge then I remembered that this criterion works for $a_n >0$ and obviously $-1<\cos(3n+1)<1$ making my approach wrong.
Could I have a hint about what criterion should I use here?

You’re actually really close. Absolute convergence implies convergence, meaning that $\sum_{n=1}^{+\infty} |a_n|$ converges $\Rightarrow \sum_{n=1}^{+\infty} a_n$ converges.
