- By hypothesis, $\sum_{n=0}^\infty a_n$ is an infinite series, and for a fixed
^{1}positive number $0 < q < 1$ there is 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. $\sqrt[n]{|a_n|}\le q$ for all $n\ge N.$ - This is equivalent to $|a_n|\le q^n$ for all $n\ge N.$
- Since the geometric series $\sum_{n=N}^\infty q_n$ is convergent, it follows from the direct comparison test for absolutely convergent series that $\sum_{n=0}^\infty a_n$ is absolutely convergent.
- Now, assume there is an index $N\in\mathbb N$ such that $\sqrt[n]{|a_n|}\ge 1$ for all $n\ge N,$ or at least $\sqrt[n]{|a_n|}\ge 1$ for infinitely many $n\in\mathbb N.$
- This is equivalent to $|a_n|\ge 1.$
- Therefore, the sequence $(a_n)_{n\in\mathbb N}$ cannot converge to zero.
- By contraposition to convergent series implies convergent sequence, the series $\sum_{n=0}^\infty a_n$ is divergent.

^{1} Please note that you have to find a fixed number $q$ that is $< 1.$ In order to be able to apply the root test, it does not suffice to show that $\sqrt[n]{|a_n|} < 1$ for all $n > N,$ the inequality must be $\sqrt[n]{|a_n|} < q < 1.$

q.e.d

| | | | created: 2020-02-09 10:10:53 | modified: 2020-02-09 10:12:57 | by: | references: [586]

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[586] **Heuser Harro**: “Lehrbuch der Analysis, Teil 1”, B.G. Teubner Stuttgart, 1994, 11. Auflage