One of the often used tests for convergence ($L\lt 1$) and divergence ($L\gt 1$) of an infinite series is the ratio test.
The idea behind it, why it works is the geometric series which dominates (or not) the tested series.
My question:
With the idea in mind that the geometric series dominates (or not) the tested one, it is a mystery to me why the test is inconclusive for the case $L=1$, because the geometric series clearly diverges in the case $x\geq 1$.
I see that there are examples for cases where $L=1$ that are convergent yet, I don't get why. I have no understanding and no intuition for that case.
Could anybody help? Thank you!