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?

Once we have a valid proof of the ratio test, then it can be applied to any series if it satisfies the hypotheses, including geometric series. It does not matter how the ratio test was proven. This is in no way circular reasoning even if the proof used results about geometric series. The idea of circular reasoning is a logical fallacy, but that is not what is being done here. We are not assuming what we are trying to prove. In fact, if we are allowed to assume statement $A$ because it is given to us, then the proof of $A$ is immediate from our given assumption. This is not circular reasoning because we are explicitly allowed to use statement $A$ which is somehow given to us to use.
The OP gave a link to course notes that state 'We cannot prove that a geometric series $\sum r^k$ is convergent/divergent by applying the Ratio Test. Why not?" That's a convincing argument, isn't it? Despite the hypotheses for "D'Alembert's Ratio Test" are satisfied by any positive geometric series.
