You made an algebraic mistake. You have the quotient $ 1/n\over 3n^2.$ If you wish to "move the numerator $1/n$ downstairs", you need to take its reciprocal, ${1\over 1/n}=n$, first and do it: $ {\color{maroon}{1/n}\over 3n^2}={1\over \color{maroon}n\cdot 3n^2}. $ You did this correctly.
However, when you "moved the denominator $3n^2$ upstairs", you failed to use its reciprocal $1/3n^2$. Done correctly, you would have obtained $ {1/n\over\color{maroon}{ 3n^2}}={(1/n)\cdot(\color{maroon}{1/3n^2})\over1 }. $
Note both methods give $1\over 3n^3$ as a result; there is no need to do both. Perhaps the simplest thing to do in order to simplify your expression is to write $ {1/n\over 3n^2}={1\over n}\cdot{1\over 3n^2}={1\over n\cdot3n^2}={1\over 3n^3}. $