To test this series for convergence

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{4^n+1}{6^n}$

You could use the Limit Comparison Test, comparing it to the series ${\displaystyle \underset{n=1}{\overset{}{}}}{r}^n$ where $r=\frac{2}{3}$

Completing the test, it shows the series:

Diverges

Converges