EDB — 21B

[21B]

• If $𝛼<1$, having fixed $L\in (𝛼,1)$ you have eventually $\sqrt[n]{|a_n|}<L$ so there is a $N$ for which $|a_n|\le L^{N-n}$ for each $n\ge N$ and we conclude by comparison with the geometric series.

• For the two series $1/n$ and $1/n^2$ you have $𝛼=1$.

• If $𝛼>1$ you have frequently $\sqrt[n]{|a_n|}>1$ So $|a_n|>1$, contrary to the necessary criterion.