Non-equivalence of D'Alembert's and Cauchy's criterion? Is there a simple example where D'Alembert's and Cauchy's criterion (the root test) for convergence of infinite series don't agree, i.e. one of them proves inconclusive?
Can you explain why that happens? Intuitive explanations along with rigorous reasoning are more than welcome!

Notice that Cauchy's root test uses $\limsup\sqrt[n]{|a_n|}$, which always exists, while d'Alembert ratio test requires $\lim\left|\frac{a_{n+1}}{a_n}\right|$ to exist. So one can find examples where Cauchy's test works, but d'Alembert doesn't. But if the the limit exists, then $$\limsup\sqrt[n]{|a_n|}=\lim\left|\frac{a_{n+1}}{a_n}\right|$$
So they both give the same result.
