Exercises: Subsequences, Accumulation points, and Cauchy sequences – "Math for Non-Geeks"

Exercises: Lim inf and Lim sup

Hint

Take a look at the inequalities proven in subtask 1. In subtask 2 even holds that $\liminf _{n\to \infty }{\frac {a_{n+1}}{a_{n}}}<\liminf _{n\to \infty }{\sqrt[{n}]{a_{n}}}<\limsup _{n\to \infty }{\sqrt[{n}]{a_{n}}}<\limsup _{n\to \infty }{\frac {a_{n+1}}{a_{n}}}$

Hint

Taking a look at the limits from subtask 4 and using reciprocals, we get: $\lim _{n\to \infty }{\sqrt[{n}]{n!}}=\infty \ {\text{ und }}\lim _{n\to \infty }{\frac {\sqrt[{n}]{n!}}{n}}={\frac {1}{e}}$