Answer
Converges
Work Step by Step
Here, we have the given series $\Sigma_{n=1}^{\infty} \dfrac{\cos (n)}{3^n}$
Next, we will use the direct comparison test for the given series to check whether it is convergent or divergent.
Let us consider that $a_n=\dfrac{\cos (n)}{3^n}$ and $b_n=\dfrac{1}{3^n}$. This implies that $a_n \leq b_n$
From the above series, we can see that the series $b_n$ shows a geometric series with common ratio $r=\dfrac{1}{3} \implies \dfrac{1}{3} \lt 1$.
Hence, we can conclude that the given series converges by the direct comparison test
