Why can't the ratio test be used for geometric series? The ratio test says that, for $a_k\neq 0$, if
$$\lim_{k\to\infty}\left|\frac{a_{k+1}}{a_k}\right|=L$$
exists, then if $0\leq L <1$, then $\sum_k a_k$ converges. If $L>1$, it diverges.
The notes I'm reading say that it's inadmissible to use the ratio test to test for convergence of a geometric series. I can't see why this should be the case. 
Say we have some geometric series $\sum_kar^k$. Then
$$\lim_{k\to\infty}\left|\frac{a_{k+1}}{a_k}\right|=\lim_{k\to\infty}\frac{\left|ar^{k+1}\right|}{\left|ar^k\right|}=|r|.$$
So the ratio test tells us that the geometric series converges for $|r|<1$, and diverges for $|r|>1$, which is exactly what we get by using the formula
$$\sum_{k=1}^n ar^k=a\left(\frac{1-r^{n+1}}{1-r}\right).$$
What is an example that demonstrates why the ratio test is inadmissible for a geometric series?

The ratio test is not inadmissible for geometric series. Its hypotheses do not exclude geometric series, therefore it applies, and its proof must support this.
One common proof structure would be:
Theorem A: Geometric series converges. Proof: direct argument.
Theorem B: Ratio test with usual hypotheses. Proof: show that this is implied by theorem A as in the answer by Xander Henderson.
Of course, the proof of Theorem A cannot use Theorem B, otherwise we have a circular argument. Undoubtedly this is what your notes are trying to say.
However, once we have a valid proof of Theorem B, it certainly applies to geometric series:
Theorem C: Geometric series converges. Proof: theorem B.
This proof of Theorem C may seem absurdly indirect: why wouldn't we just cite Theorem A? Well, consider:
Theorem D: some other theorem whose hypotheses imply those of the ratio test. Proof: Theorem B.
It would be very annoying, and more importantly unnecessary, to instead write:
Theorem D: some other theorem whose hypotheses imply those of the ratio test. Proof: If the series is geometric, then see Theorem A. Otherwise Theorem B applies.
