Suppose $\omega$ scales as $t^{n}$, where $t$ is a constant and $n \in \Bbb{Z}^{+}$.
In order to show it graphically, we can consider the ratio $\frac{\omega_{n+1}}{\omega_{n}}$ and it is equal to the constant $t$. So when we plot $\frac{\omega_{n+1}}{\omega_{n}}$ vs $n$ it gives us a sequence of points converging to the straight line parallel to the $n$-axis (joining the discrete points).
But this gets tricky if $\omega$ scales as $\frac{t^{n+1}}{n}$ as then $\frac{\omega_{n+1}}{\omega_{n}}$ equals something not a constant and hence not easy to interpret or prove graphically. Any ideas?
Why not look at $n\omega_n\sim t^n$ and thus
$$\frac{(n+1)\omega_{n+1}}{n\omega_n}\to t$$
which should tend to a straight line as well.