Proof
If \(๐ผ{\lt}1\), having fixed \(Lโ(๐ผ,1)\) you have eventually \(\sqrt[n]{|a_{n}|}{\lt}L\) so there is a \(N\) for which \(|a_ n|โค L^{N-n}\) for each \(nโฅ N\) and we conclude by comparison with the geometric series.
For the two series \(1/n\) and \(1/n^ 2\) you have \(๐ผ=1\).
If \(๐ผ{\gt}1\) you have frequently \(\sqrt[n]{|a_{n}|}{\gt}1\) So \(|a_ n|{\gt}1\), contrary to the necessary criterion.