Do we need to use the Ratio/Root test to determine divergence of a series?
A series like this perhaps:
$$\sum_{n=1}^\infty \frac {3^n n!}{n^n}$$
Although the limit of this sequence is indeed not zero, I don't think most Calc I or II students would be able to prove it easily without resorting to a very tailored approach for this problem. On the other hand, the ratio test handles this one easily.
That is, provided they are not commonly aware that $$\lim_{n\to\infty} \frac {(n!)^{\frac 1 n}} n=\frac 1 e$$ (I wasn't when I took Calc I and II.)