Proposition: Root Test

Let $\sum_{n=0}^\infty a_n$ be an infinite series such that for a given positive number $0 < q < 1$ there exists an index $N$ such that for all $n\ge N$ the n-th root of absolute values is smaller than or equal $q$, i.e. $$\sqrt[n]{|a_n|}\le q\quad\forall n\ge N,$$ then the infinite series $\sum_{n=0}^\infty a_n$ is absolutely convergent. But if there is an index $N\in\mathbb N$ such that $$\sqrt[n]{|a_n|}\ge 1\quad\forall n\ge N,$$ or at least $\sqrt[n]{|a_n|}\ge 1$ for infinitely many $n\in\mathbb N,$ then $\sum_{n=0}^\infty a_n$ is divergent.

Proofs: 1

Proofs: 1

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