whether the adhering to collection merge or deviate

Your collection can be created as $$\sum_{n=1}^\infty \frac {3}{6^n}.$$ It is a geometric collection with $a=3/6=1/2, r=1/6$.

You can make use of the comparison test :

\[\sum_{n=1}^\infty \frac{1}{2^n 3^{n-1}}<\sum_{n=1}^\infty \frac{1}{2^n}=1.\]