Answer
Converges
Work Step by Step
Since, we have $\Sigma_{n=1}^\infty \dfrac{1}{3^{n-1}+1} \lt \Sigma_{n=1}^\infty\dfrac{1}{3^{n-1}}$
Here, $\Sigma_{n=1}^\infty\dfrac{1}{3^{n-1}}$ is a geometric convergent series with common ratio $r=\dfrac{1}{3} \gt 1$
Thus the series converges by the direct comparison test.
