The inequality $\limsup \frac{c_{n+1}}{c_n} \geqslant \limsup \sqrt[n]{c_n}$
Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$