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.
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!