Proposition: Limit Test for Roots or Ratios

If either the sequence of n-th roots $(\sqrt[n]{|a_n|})_{n\in\mathbb N},$ or the sequence of ratios $\left(\frac{|a_{n+1}|}{|a_n|}\right)_{n\in\mathbb N}$ is convergent to a limit $\alpha > 0,$ then the infinite series $\sum_{n=0}^\infty a_n$ is convergent if $0 < \alpha < 1,$ and divergent, if $\alpha > 1.$

Note: In the case $\alpha=1,$ this criterion allows no conclusion whatsoever on the convergence behavior of the series $\sum_{n=0}^\infty a_n,$ and further investigation is needed!

Proofs: 1

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  1. Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition